Properties

Degree 2
Conductor $ 5 \cdot 29 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 5-s − 2·7-s + 3·8-s − 3·9-s + 10-s − 6·11-s + 2·13-s + 2·14-s − 16-s − 2·17-s + 3·18-s − 2·19-s + 20-s + 6·22-s + 2·23-s + 25-s − 2·26-s + 2·28-s − 29-s + 2·31-s − 5·32-s + 2·34-s + 2·35-s + 3·36-s + 10·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.755·7-s + 1.06·8-s − 9-s + 0.316·10-s − 1.80·11-s + 0.554·13-s + 0.534·14-s − 1/4·16-s − 0.485·17-s + 0.707·18-s − 0.458·19-s + 0.223·20-s + 1.27·22-s + 0.417·23-s + 1/5·25-s − 0.392·26-s + 0.377·28-s − 0.185·29-s + 0.359·31-s − 0.883·32-s + 0.342·34-s + 0.338·35-s + 1/2·36-s + 1.64·37-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(145\)    =    \(5 \cdot 29\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{145} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 145,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;29\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 + T \)
29 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−19.70788026644800, −18.95783529205681, −18.22854134158185, −17.41755572679201, −16.39121880692705, −15.73850649233830, −14.54646000355637, −13.30559077245382, −12.87468148069786, −11.24854153702415, −10.50387677609449, −9.378870263890314, −8.402368279712544, −7.649471925073914, −6.019568004660478, −4.649718277081199, −2.952031825994031, 0, 2.952031825994031, 4.649718277081199, 6.019568004660478, 7.649471925073914, 8.402368279712544, 9.378870263890314, 10.50387677609449, 11.24854153702415, 12.87468148069786, 13.30559077245382, 14.54646000355637, 15.73850649233830, 16.39121880692705, 17.41755572679201, 18.22854134158185, 18.95783529205681, 19.70788026644800

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