Properties

Degree 2
Conductor $ 2 \cdot 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  + 2-s + 4-s − 5-s − 7-s + 8-s − 3·9-s − 10-s + 4·11-s − 6·13-s − 14-s + 16-s + 2·17-s − 3·18-s − 20-s + 4·22-s + 25-s − 6·26-s − 28-s + 6·29-s + 8·31-s + 32-s + 2·34-s + 35-s − 3·36-s − 10·37-s − 40-s + 2·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 9-s − 0.316·10-s + 1.20·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.707·18-s − 0.223·20-s + 0.852·22-s + 1/5·25-s − 1.17·26-s − 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.342·34-s + 0.169·35-s − 1/2·36-s − 1.64·37-s − 0.158·40-s + 0.312·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(70\)    =    \(2 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{70} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 70,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.180304346$
$L(\frac12)$  $\approx$  $1.180304346$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 + T \)
good3 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−19.53188599188130, −19.38819936835090, −17.40149380001984, −16.81984676209900, −15.56000405753534, −14.53586643730421, −13.89417772743579, −12.14707369305425, −11.96572717133546, −10.37599431086571, −8.977646092372606, −7.454202487299858, −6.176161440923152, −4.639233419499348, −3.003813725090910, 3.003813725090910, 4.639233419499348, 6.176161440923152, 7.454202487299858, 8.977646092372606, 10.37599431086571, 11.96572717133546, 12.14707369305425, 13.89417772743579, 14.53586643730421, 15.56000405753534, 16.81984676209900, 17.40149380001984, 19.38819936835090, 19.53188599188130

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