Properties

Label 8-2352e4-1.1-c2e4-0-2
Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Analytic cond. $1.68690\times 10^{7}$
Root an. cond. $8.00545$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 6·9-s − 26·13-s + 40·17-s − 69·25-s − 2·29-s + 38·37-s + 4·41-s − 12·45-s − 338·53-s + 192·61-s − 52·65-s − 382·73-s + 27·81-s + 80·85-s + 244·89-s − 50·97-s + 208·101-s − 490·109-s − 500·113-s + 156·117-s + 125·121-s − 162·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 2/5·5-s − 2/3·9-s − 2·13-s + 2.35·17-s − 2.75·25-s − 0.0689·29-s + 1.02·37-s + 4/41·41-s − 0.266·45-s − 6.37·53-s + 3.14·61-s − 4/5·65-s − 5.23·73-s + 1/3·81-s + 0.941·85-s + 2.74·89-s − 0.515·97-s + 2.05·101-s − 4.49·109-s − 4.42·113-s + 4/3·117-s + 1.03·121-s − 1.29·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.68690\times 10^{7}\)
Root analytic conductor: \(8.00545\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2294941072\)
\(L(\frac12)\) \(\approx\) \(0.2294941072\)
\(L(2)\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( ( 1 - T + 36 T^{2} - p^{2} T^{3} + p^{4} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 125 T^{2} + 4332 T^{4} - 125 p^{4} T^{6} + p^{8} T^{8} \)
13$D_{4}$ \( ( 1 + p T + 366 T^{2} + p^{3} T^{3} + p^{4} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 - 20 T + 450 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 + 175 T^{2} + 246624 T^{4} + 175 p^{4} T^{6} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 - 1940 T^{2} + 1496934 T^{4} - 1940 p^{4} T^{6} + p^{8} T^{8} \)
29$D_{4}$ \( ( 1 + T - 42 T^{2} + p^{2} T^{3} + p^{4} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 1126 T^{2} + 317211 T^{4} - 1126 p^{4} T^{6} + p^{8} T^{8} \)
37$D_{4}$ \( ( 1 - 19 T + 2130 T^{2} - 19 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 2 T + 1938 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 3557 T^{2} + 9145308 T^{4} - 3557 p^{4} T^{6} + p^{8} T^{8} \)
47$C_2^2$ \( ( 1 - 3830 T^{2} + p^{4} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 + 169 T + 234 p T^{2} + 169 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 11057 T^{2} + 54465456 T^{4} - 11057 p^{4} T^{6} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 - 96 T + 9518 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 17173 T^{2} + 114019836 T^{4} - 17173 p^{4} T^{6} + p^{8} T^{8} \)
71$D_4\times C_2$ \( 1 - 4748 T^{2} + 53176038 T^{4} - 4748 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 + 191 T + 19764 T^{2} + 191 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 17182 T^{2} + 137226531 T^{4} - 17182 p^{4} T^{6} + p^{8} T^{8} \)
83$D_4\times C_2$ \( 1 - 1997 T^{2} - 7277460 T^{4} - 1997 p^{4} T^{6} + p^{8} T^{8} \)
89$D_{4}$ \( ( 1 - 122 T + 19506 T^{2} - 122 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 25 T + 5280 T^{2} + 25 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.10918364930466779244281482523, −6.06682682705351460743788887641, −5.78914658202848210962390338862, −5.43437660026085986064011546305, −5.38276061030056758203819579296, −5.23921231972086812958045825196, −5.11227501335075410962004051213, −4.66822415348065084963917428030, −4.43797930298953191348345294251, −4.33943859151869664147360559493, −4.24966348661247150995959718664, −3.67411677134004695245509184099, −3.39602622685667917950425102277, −3.36027950144457196181857590950, −3.34305405483823786604794591535, −2.77780370484327199790217088809, −2.44182202923554376284535595474, −2.40943960575313565103188518209, −2.38599473418893607404443536566, −1.56825858263192121563594084852, −1.56446804686825234270476135346, −1.39791534401358633024276037257, −1.01008058783321025410811582580, −0.30851031748906938926625611512, −0.097150150796574089572048672953, 0.097150150796574089572048672953, 0.30851031748906938926625611512, 1.01008058783321025410811582580, 1.39791534401358633024276037257, 1.56446804686825234270476135346, 1.56825858263192121563594084852, 2.38599473418893607404443536566, 2.40943960575313565103188518209, 2.44182202923554376284535595474, 2.77780370484327199790217088809, 3.34305405483823786604794591535, 3.36027950144457196181857590950, 3.39602622685667917950425102277, 3.67411677134004695245509184099, 4.24966348661247150995959718664, 4.33943859151869664147360559493, 4.43797930298953191348345294251, 4.66822415348065084963917428030, 5.11227501335075410962004051213, 5.23921231972086812958045825196, 5.38276061030056758203819579296, 5.43437660026085986064011546305, 5.78914658202848210962390338862, 6.06682682705351460743788887641, 6.10918364930466779244281482523

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