Properties

Degree 2
Conductor $ 5 \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s − 5-s + 7-s − 2·9-s − 3·11-s − 2·12-s + 5·13-s − 15-s + 4·16-s + 3·17-s + 2·19-s + 2·20-s + 21-s − 6·23-s + 25-s − 5·27-s − 2·28-s + 3·29-s − 4·31-s − 3·33-s − 35-s + 4·36-s + 2·37-s + 5·39-s − 12·41-s − 10·43-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s − 2/3·9-s − 0.904·11-s − 0.577·12-s + 1.38·13-s − 0.258·15-s + 16-s + 0.727·17-s + 0.458·19-s + 0.447·20-s + 0.218·21-s − 1.25·23-s + 1/5·25-s − 0.962·27-s − 0.377·28-s + 0.557·29-s − 0.718·31-s − 0.522·33-s − 0.169·35-s + 2/3·36-s + 0.328·37-s + 0.800·39-s − 1.87·41-s − 1.52·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(35\)    =    \(5 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{35} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 35,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7029112391$
$L(\frac12)$  $\approx$  $0.7029112391$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 + T \)
7 \( 1 - T \)
good2 \( 1 + p T^{2} \)
3 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.97965963083303, −18.64374751791110, −18.06224800294894, −16.59564054902116, −15.27252635693957, −14.09621015074921, −13.34832607225341, −11.80747475770718, −10.26175613408034, −8.679498853252053, −7.980947296320423, −5.479075640409833, −3.616890030455143, 3.616890030455143, 5.479075640409833, 7.980947296320423, 8.679498853252053, 10.26175613408034, 11.80747475770718, 13.34832607225341, 14.09621015074921, 15.27252635693957, 16.59564054902116, 18.06224800294894, 18.64374751791110, 19.97965963083303

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