Properties

Degree 2
Conductor $ 2 \cdot 31 $
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 − 2·5-s + 8-s − 3·9-s − 2·10-s + 2·13-s + 16-s − 6·17-s − 3·18-s + 4·19-s − 2·20-s + 8·23-s − 25-s + 2·26-s + 2·29-s − 31-s + 32-s − 6·34-s − 3·36-s + 10·37-s + 4·38-s − 2·40-s − 6·41-s + 8·43-s + 6·45-s + 8·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 9-s − 0.632·10-s + 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.917·19-s − 0.447·20-s + 1.66·23-s − 1/5·25-s + 0.392·26-s + 0.371·29-s − 0.179·31-s + 0.176·32-s − 1.02·34-s − 1/2·36-s + 1.64·37-s + 0.648·38-s − 0.316·40-s − 0.937·41-s + 1.21·43-s + 0.894·45-s + 1.17·46-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(62\)    =    \(2 \cdot 31\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{62} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 62,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.10874$
$L(\frac12)$  $\approx$  $1.10874$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−15.05691750776431447573088356666, −13.90986476156025600386975635688, −12.88268162615932246882266504036, −11.52935425193863190686584011034, −11.06274569152331522479839207162, −9.056498267458590117851162621111, −7.76857292337562945056886541715, −6.32858579420662158347624019611, −4.74203912193695697715499891246, −3.16188554299090604324575557253, 3.16188554299090604324575557253, 4.74203912193695697715499891246, 6.32858579420662158347624019611, 7.76857292337562945056886541715, 9.056498267458590117851162621111, 11.06274569152331522479839207162, 11.52935425193863190686584011034, 12.88268162615932246882266504036, 13.90986476156025600386975635688, 15.05691750776431447573088356666

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