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Properties

Label 10-4020e5-1.1-c1e5-0-0
Degree $10$
Conductor $1.050\times 10^{18}$
Sign $-1$
Analytic cond. $3.40812\times 10^{7}$
Root an. cond. $5.66567$
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  − 5·3-s + 5·5-s − 3·7-s + 15·9-s + 3·11-s − 11·13-s − 25·15-s − 11·17-s + 6·19-s + 15·21-s + 3·23-s + 15·25-s − 35·27-s − 2·29-s + 31-s − 15·33-s − 15·35-s − 13·37-s + 55·39-s − 13·41-s − 9·43-s + 75·45-s − 12·47-s − 14·49-s + 55·51-s − 19·53-s + 15·55-s + ⋯
L(s)  = 1  − 2.88·3-s + 2.23·5-s − 1.13·7-s + 5·9-s + 0.904·11-s − 3.05·13-s − 6.45·15-s − 2.66·17-s + 1.37·19-s + 3.27·21-s + 0.625·23-s + 3·25-s − 6.73·27-s − 0.371·29-s + 0.179·31-s − 2.61·33-s − 2.53·35-s − 2.13·37-s + 8.80·39-s − 2.03·41-s − 1.37·43-s + 11.1·45-s − 1.75·47-s − 2·49-s + 7.70·51-s − 2.60·53-s + 2.02·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{5} \cdot 5^{5} \cdot 67^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{10} \cdot 3^{5} \cdot 5^{5} \cdot 67^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{10} \cdot 3^{5} \cdot 5^{5} \cdot 67^{5}\)
Sign: $-1$
Analytic conductor: \(3.40812\times 10^{7}\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{10} \cdot 3^{5} \cdot 5^{5} \cdot 67^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + T )^{5} \)
5$C_1$ \( ( 1 - T )^{5} \)
67$C_1$ \( ( 1 + T )^{5} \)
good7$C_2 \wr S_5$ \( 1 + 3 T + 23 T^{2} + 68 T^{3} + 241 T^{4} + 660 T^{5} + 241 p T^{6} + 68 p^{2} T^{7} + 23 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 - 3 T + 29 T^{2} - 60 T^{3} + 35 p T^{4} - 712 T^{5} + 35 p^{2} T^{6} - 60 p^{2} T^{7} + 29 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 11 T + 102 T^{2} + 599 T^{3} + 3098 T^{4} + 11862 T^{5} + 3098 p T^{6} + 599 p^{2} T^{7} + 102 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 11 T + 115 T^{2} + 695 T^{3} + 4088 T^{4} + 16919 T^{5} + 4088 p T^{6} + 695 p^{2} T^{7} + 115 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 - 6 T + 43 T^{2} - 258 T^{3} + 1470 T^{4} - 5525 T^{5} + 1470 p T^{6} - 258 p^{2} T^{7} + 43 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 3 T + 37 T^{2} - 98 T^{3} + 417 T^{4} - 2453 T^{5} + 417 p T^{6} - 98 p^{2} T^{7} + 37 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 2 T + 45 T^{2} - 66 T^{3} + 2142 T^{4} + 1785 T^{5} + 2142 p T^{6} - 66 p^{2} T^{7} + 45 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - T + 75 T^{2} + 328 T^{3} + 1275 T^{4} + 22862 T^{5} + 1275 p T^{6} + 328 p^{2} T^{7} + 75 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 13 T + 135 T^{2} + 922 T^{3} + 5933 T^{4} + 31335 T^{5} + 5933 p T^{6} + 922 p^{2} T^{7} + 135 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 13 T + 207 T^{2} + 38 p T^{3} + 14601 T^{4} + 81272 T^{5} + 14601 p T^{6} + 38 p^{3} T^{7} + 207 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 9 T + 53 T^{2} + 120 T^{3} + 2299 T^{4} + 15920 T^{5} + 2299 p T^{6} + 120 p^{2} T^{7} + 53 p^{3} T^{8} + 9 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 12 T + 221 T^{2} + 2144 T^{3} + 20300 T^{4} + 148487 T^{5} + 20300 p T^{6} + 2144 p^{2} T^{7} + 221 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 19 T + 228 T^{2} + 1171 T^{3} + 1956 T^{4} - 488 p T^{5} + 1956 p T^{6} + 1171 p^{2} T^{7} + 228 p^{3} T^{8} + 19 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 6 T + 127 T^{2} - 562 T^{3} + 9064 T^{4} - 21293 T^{5} + 9064 p T^{6} - 562 p^{2} T^{7} + 127 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 3 T + 238 T^{2} + 663 T^{3} + 25704 T^{4} + 59788 T^{5} + 25704 p T^{6} + 663 p^{2} T^{7} + 238 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 6 T + 311 T^{2} - 1489 T^{3} + 41597 T^{4} - 152470 T^{5} + 41597 p T^{6} - 1489 p^{2} T^{7} + 311 p^{3} T^{8} - 6 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 21 T + 465 T^{2} + 5845 T^{3} + 72518 T^{4} + 625715 T^{5} + 72518 p T^{6} + 5845 p^{2} T^{7} + 465 p^{3} T^{8} + 21 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + T + 23 T^{2} - 362 T^{3} + 4893 T^{4} - 32514 T^{5} + 4893 p T^{6} - 362 p^{2} T^{7} + 23 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 8 T + 111 T^{2} + 377 T^{3} + 14981 T^{4} + 112874 T^{5} + 14981 p T^{6} + 377 p^{2} T^{7} + 111 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 29 T + 747 T^{2} + 11591 T^{3} + 161040 T^{4} + 1600867 T^{5} + 161040 p T^{6} + 11591 p^{2} T^{7} + 747 p^{3} T^{8} + 29 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 13 T + 172 T^{2} + 1289 T^{3} + 26352 T^{4} + 245878 T^{5} + 26352 p T^{6} + 1289 p^{2} T^{7} + 172 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.43384395731943298222452690168, −5.22557063152331660654879505428, −5.07744322067557247815078601241, −4.98292889731732890721537404827, −4.87599294999191166467191312425, −4.74499860961241544582926846339, −4.59344575301409832153579884770, −4.44474151737620504083229664768, −4.25669042746221372007407980732, −4.24338849498077599653239029906, −3.61565088194441640044875780858, −3.51675200390176891100692193003, −3.43442832410091682788545273589, −3.28584366723292151347069881303, −3.13131039281380981340023251620, −2.67264093593231255249595307296, −2.62524607125788629564206969133, −2.32769988278389236408018676709, −2.29097604605577277481109432376, −2.16232974100186983645419391827, −1.56666661502085449972045843761, −1.43905154719990733739667551325, −1.43365664974333987350102799626, −1.31576975207790599924972908937, −1.22746292667416931846844606418, 0, 0, 0, 0, 0, 1.22746292667416931846844606418, 1.31576975207790599924972908937, 1.43365664974333987350102799626, 1.43905154719990733739667551325, 1.56666661502085449972045843761, 2.16232974100186983645419391827, 2.29097604605577277481109432376, 2.32769988278389236408018676709, 2.62524607125788629564206969133, 2.67264093593231255249595307296, 3.13131039281380981340023251620, 3.28584366723292151347069881303, 3.43442832410091682788545273589, 3.51675200390176891100692193003, 3.61565088194441640044875780858, 4.24338849498077599653239029906, 4.25669042746221372007407980732, 4.44474151737620504083229664768, 4.59344575301409832153579884770, 4.74499860961241544582926846339, 4.87599294999191166467191312425, 4.98292889731732890721537404827, 5.07744322067557247815078601241, 5.22557063152331660654879505428, 5.43384395731943298222452690168

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