Properties

Degree 2
Conductor $ 2 \cdot 31 $
Sign $0.780 - 0.624i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)2-s + (−1.02 + 3.15i)3-s + (−0.809 + 0.587i)4-s + 2.66·5-s + 3.31·6-s + (−0.5 + 0.363i)7-s + (0.809 + 0.587i)8-s + (−6.45 − 4.69i)9-s + (−0.823 − 2.53i)10-s + (1.09 − 0.794i)11-s + (−1.02 − 3.15i)12-s + (0.729 − 2.24i)13-s + (0.5 + 0.363i)14-s + (−2.72 + 8.40i)15-s + (0.309 − 0.951i)16-s + (2.15 + 1.56i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.591 + 1.81i)3-s + (−0.404 + 0.293i)4-s + 1.19·5-s + 1.35·6-s + (−0.188 + 0.137i)7-s + (0.286 + 0.207i)8-s + (−2.15 − 1.56i)9-s + (−0.260 − 0.801i)10-s + (0.329 − 0.239i)11-s + (−0.295 − 0.909i)12-s + (0.202 − 0.622i)13-s + (0.133 + 0.0970i)14-s + (−0.704 + 2.16i)15-s + (0.0772 − 0.237i)16-s + (0.523 + 0.380i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 - 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.780 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(62\)    =    \(2 \cdot 31\)
\( \varepsilon \)  =  $0.780 - 0.624i$
motivic weight  =  \(1\)
character  :  $\chi_{62} (35, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 62,\ (\ :1/2),\ 0.780 - 0.624i)$
$L(1)$  $\approx$  $0.703652 + 0.246866i$
$L(\frac12)$  $\approx$  $0.703652 + 0.246866i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;31\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (5.42 + 1.26i)T \)
good3 \( 1 + (1.02 - 3.15i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 - 2.66T + 5T^{2} \)
7 \( 1 + (0.5 - 0.363i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-1.09 + 0.794i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.729 + 2.24i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.15 - 1.56i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.98 + 6.09i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-2.37 - 1.72i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.17 - 3.62i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + 4.57T + 37T^{2} \)
41 \( 1 + (0.676 + 2.08i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-1.92 - 5.93i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (1.5 - 4.61i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.78 + 2.02i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.42 + 10.5i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + 2.39T + 61T^{2} \)
67 \( 1 + 9.40T + 67T^{2} \)
71 \( 1 + (6.62 + 4.81i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.47 + 2.52i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.64 - 1.91i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.354 + 1.08i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (6.85 - 4.98i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-2.42 + 1.76i)T + (29.9 - 92.2i)T^{2} \)
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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

−15.23612411586198716042624535421, −14.15136584193962658896336350922, −12.77340118398644771838113325149, −11.25935508714175524717406010206, −10.54231100778111091215952717236, −9.580925548944279506656268402387, −8.906805851525785977659568773206, −6.02729670419625671369950665219, −4.92051164826359304092076805227, −3.25657413500718852002690975402, 1.74554332547384636926130650134, 5.57341494907444879961472610434, 6.40721647464635013961024553604, 7.36600666878995911979968219896, 8.742192915670178633708992873828, 10.26657057317951294558754532546, 11.85059406007025000204052552386, 12.89366807116952255256232287314, 13.77252495240211716704911088058, 14.45961851026902772115026016041

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