Properties

Degree $8$
Conductor $1.216\times 10^{12}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 4·7-s + 3·16-s − 8·28-s + 8·37-s − 16·43-s − 2·49-s − 4·64-s − 32·67-s − 40·79-s − 9·81-s + 40·109-s + 12·112-s + 44·121-s + 127-s + 131-s + 137-s + 139-s − 16·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 32·172-s + 173-s + ⋯
L(s)  = 1  − 4-s + 1.51·7-s + 3/4·16-s − 1.51·28-s + 1.31·37-s − 2.43·43-s − 2/7·49-s − 1/2·64-s − 3.90·67-s − 4.50·79-s − 81-s + 3.83·109-s + 1.13·112-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.31·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.07·169-s + 2.43·172-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1050} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.204383746\)
\(L(\frac12)\) \(\approx\) \(1.204383746\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
3$C_2^2$ \( 1 + p^{2} T^{4} \)
5 \( 1 \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
good11$C_2$ \( ( 1 - p T^{2} )^{4} \)
13$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - p T^{2} )^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{4} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 160 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{4} \)
97$C_2^2$ \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \)
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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

−7.13212304164963986819618312753, −6.89799243615811870020358748348, −6.84498733593176397322013507869, −6.18297487871858723343222126833, −6.11871470584013594334326450483, −6.01828880890777802479482610364, −5.80037836963358144916291297199, −5.34889516316225397284979613699, −5.33583705371475044487665834192, −4.92519658321201342349905784608, −4.71388457137282644029072364478, −4.58833020679123276816062157146, −4.41928024822430936023081206773, −4.11181662973054219296095248027, −4.02733402092110112665028601119, −3.38507631863004167325127807573, −3.28357244760207935136789229141, −2.91728346294052171119198843118, −2.91252501036235362599200584448, −2.30483622545207201596114306436, −1.83621883211207400100821098209, −1.55682359180407164960799511877, −1.53570434389176955491284794631, −0.925177338972005694605538232554, −0.27259851216497968689138301265, 0.27259851216497968689138301265, 0.925177338972005694605538232554, 1.53570434389176955491284794631, 1.55682359180407164960799511877, 1.83621883211207400100821098209, 2.30483622545207201596114306436, 2.91252501036235362599200584448, 2.91728346294052171119198843118, 3.28357244760207935136789229141, 3.38507631863004167325127807573, 4.02733402092110112665028601119, 4.11181662973054219296095248027, 4.41928024822430936023081206773, 4.58833020679123276816062157146, 4.71388457137282644029072364478, 4.92519658321201342349905784608, 5.33583705371475044487665834192, 5.34889516316225397284979613699, 5.80037836963358144916291297199, 6.01828880890777802479482610364, 6.11871470584013594334326450483, 6.18297487871858723343222126833, 6.84498733593176397322013507869, 6.89799243615811870020358748348, 7.13212304164963986819618312753

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