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Properties

Label 10-4235e5-1.1-c1e5-0-5
Degree $10$
Conductor $1.362\times 10^{18}$
Sign $-1$
Analytic cond. $4.42234\times 10^{7}$
Root an. cond. $5.81520$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $5$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 3·4-s − 5·5-s + 5·7-s − 4·8-s + 6·12-s − 10·13-s + 10·15-s + 5·16-s − 2·17-s + 3·19-s + 15·20-s − 10·21-s + 3·23-s + 8·24-s + 15·25-s + 27-s − 15·28-s − 29-s − 12·31-s + 11·32-s − 25·35-s + 20·39-s + 20·40-s + 11·41-s − 10·43-s + 16·47-s + ⋯
L(s)  = 1  − 1.15·3-s − 3/2·4-s − 2.23·5-s + 1.88·7-s − 1.41·8-s + 1.73·12-s − 2.77·13-s + 2.58·15-s + 5/4·16-s − 0.485·17-s + 0.688·19-s + 3.35·20-s − 2.18·21-s + 0.625·23-s + 1.63·24-s + 3·25-s + 0.192·27-s − 2.83·28-s − 0.185·29-s − 2.15·31-s + 1.94·32-s − 4.22·35-s + 3.20·39-s + 3.16·40-s + 1.71·41-s − 1.52·43-s + 2.33·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 7^{5} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 7^{5} \cdot 11^{10}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(5^{5} \cdot 7^{5} \cdot 11^{10}\)
Sign: $-1$
Analytic conductor: \(4.42234\times 10^{7}\)
Root analytic conductor: \(5.81520\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 5^{5} \cdot 7^{5} \cdot 11^{10} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad5$C_1$ \( ( 1 + T )^{5} \)
7$C_1$ \( ( 1 - T )^{5} \)
11 \( 1 \)
good2$C_2 \wr S_5$ \( 1 + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} + 13 T^{5} + p^{3} T^{6} + p^{4} T^{7} + 3 p^{3} T^{8} + p^{5} T^{10} \)
3$C_2 \wr S_5$ \( 1 + 2 T + 4 T^{2} + 7 T^{3} + 2 p^{2} T^{4} + 25 T^{5} + 2 p^{3} T^{6} + 7 p^{2} T^{7} + 4 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 10 T + 73 T^{2} + 421 T^{3} + 1998 T^{4} + 7699 T^{5} + 1998 p T^{6} + 421 p^{2} T^{7} + 73 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 2 T + 57 T^{2} + 93 T^{3} + 1546 T^{4} + 1997 T^{5} + 1546 p T^{6} + 93 p^{2} T^{7} + 57 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 3 T + 44 T^{2} - 6 p T^{3} + 1142 T^{4} - 1897 T^{5} + 1142 p T^{6} - 6 p^{3} T^{7} + 44 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 3 T + 45 T^{2} - 67 T^{3} + 1651 T^{4} - 3481 T^{5} + 1651 p T^{6} - 67 p^{2} T^{7} + 45 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + T + 109 T^{2} + T^{3} + 5251 T^{4} - 1477 T^{5} + 5251 p T^{6} + p^{2} T^{7} + 109 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 12 T + 116 T^{2} + 453 T^{3} + 1616 T^{4} - 139 T^{5} + 1616 p T^{6} + 453 p^{2} T^{7} + 116 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 40 T^{2} - 243 T^{3} + 2144 T^{4} - 6437 T^{5} + 2144 p T^{6} - 243 p^{2} T^{7} + 40 p^{3} T^{8} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 - 11 T + 175 T^{2} - 1223 T^{3} + 11437 T^{4} - 61879 T^{5} + 11437 p T^{6} - 1223 p^{2} T^{7} + 175 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 10 T + 218 T^{2} + 1473 T^{3} + 18040 T^{4} + 88937 T^{5} + 18040 p T^{6} + 1473 p^{2} T^{7} + 218 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 - 16 T + 256 T^{2} - 2458 T^{3} + 23647 T^{4} - 161699 T^{5} + 23647 p T^{6} - 2458 p^{2} T^{7} + 256 p^{3} T^{8} - 16 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 4 T + 98 T^{2} - 193 T^{3} + 4696 T^{4} - 293 T^{5} + 4696 p T^{6} - 193 p^{2} T^{7} + 98 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 + 32 T + 675 T^{2} + 9597 T^{3} + 106822 T^{4} + 912929 T^{5} + 106822 p T^{6} + 9597 p^{2} T^{7} + 675 p^{3} T^{8} + 32 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 40 T + 918 T^{2} + 14160 T^{3} + 163483 T^{4} + 1443777 T^{5} + 163483 p T^{6} + 14160 p^{2} T^{7} + 918 p^{3} T^{8} + 40 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 7 T + 218 T^{2} - 1484 T^{3} + 24806 T^{4} - 136015 T^{5} + 24806 p T^{6} - 1484 p^{2} T^{7} + 218 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 10 T + 320 T^{2} - 2293 T^{3} + 42034 T^{4} - 225941 T^{5} + 42034 p T^{6} - 2293 p^{2} T^{7} + 320 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 11 T + 224 T^{2} + 2294 T^{3} + 23764 T^{4} + 217565 T^{5} + 23764 p T^{6} + 2294 p^{2} T^{7} + 224 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 13 T + 300 T^{2} + 2988 T^{3} + 40636 T^{4} + 309897 T^{5} + 40636 p T^{6} + 2988 p^{2} T^{7} + 300 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 26 T + 531 T^{2} + 6951 T^{3} + 83026 T^{4} + 772193 T^{5} + 83026 p T^{6} + 6951 p^{2} T^{7} + 531 p^{3} T^{8} + 26 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 - 5 T + 350 T^{2} - 1403 T^{3} + 55471 T^{4} - 171451 T^{5} + 55471 p T^{6} - 1403 p^{2} T^{7} + 350 p^{3} T^{8} - 5 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 5 T + 362 T^{2} + 1189 T^{3} + 58577 T^{4} + 136625 T^{5} + 58577 p T^{6} + 1189 p^{2} T^{7} + 362 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.40228027001684911541461710758, −5.07238780692023546950265618168, −4.92731712964669238710548611981, −4.91346956072138449106986103680, −4.68058122992059830029739485712, −4.48914861346375483746686056716, −4.47053574540703709676938498687, −4.46897731621169775459569462581, −4.17024520988080786533029659948, −4.10398746161757936347970932605, −3.73607862712437498669193997138, −3.52857733101018879474506265543, −3.41926448975827386826530331147, −3.40281573771736381210255959023, −3.01643043278948826200283704129, −2.76589504566288891801587903061, −2.76554758369001870681166591513, −2.58090020559562173010432487331, −2.37932799084057862861877294386, −2.02432368716458933230669943131, −1.84910683627000440871186032933, −1.33819516686197425488857813623, −1.33389879330898487803248102556, −1.15056145230138208375398836377, −0.867792170165655869345317194162, 0, 0, 0, 0, 0, 0.867792170165655869345317194162, 1.15056145230138208375398836377, 1.33389879330898487803248102556, 1.33819516686197425488857813623, 1.84910683627000440871186032933, 2.02432368716458933230669943131, 2.37932799084057862861877294386, 2.58090020559562173010432487331, 2.76554758369001870681166591513, 2.76589504566288891801587903061, 3.01643043278948826200283704129, 3.40281573771736381210255959023, 3.41926448975827386826530331147, 3.52857733101018879474506265543, 3.73607862712437498669193997138, 4.10398746161757936347970932605, 4.17024520988080786533029659948, 4.46897731621169775459569462581, 4.47053574540703709676938498687, 4.48914861346375483746686056716, 4.68058122992059830029739485712, 4.91346956072138449106986103680, 4.92731712964669238710548611981, 5.07238780692023546950265618168, 5.40228027001684911541461710758

Graph of the $Z$-function along the critical line