Properties

Degree 2
Conductor $ 2 \cdot 31 $
Sign $0.659 - 0.751i$
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 + (−0.413 − 0.459i)3-s + (−0.809 + 0.587i)4-s + (1.78 + 3.09i)5-s + (0.309 − 0.535i)6-s + (−0.309 − 2.94i)7-s + (−0.809 − 0.587i)8-s + (0.273 − 2.60i)9-s + (−2.39 + 2.65i)10-s + (−2.16 − 0.965i)11-s + (0.604 + 0.128i)12-s + (−5.53 + 1.17i)13-s + (2.70 − 1.20i)14-s + (0.682 − 2.10i)15-s + (0.309 − 0.951i)16-s + (2.83 − 1.26i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.238 − 0.265i)3-s + (−0.404 + 0.293i)4-s + (0.799 + 1.38i)5-s + (0.126 − 0.218i)6-s + (−0.116 − 1.11i)7-s + (−0.286 − 0.207i)8-s + (0.0912 − 0.867i)9-s + (−0.756 + 0.839i)10-s + (−0.654 − 0.291i)11-s + (0.174 + 0.0370i)12-s + (−1.53 + 0.326i)13-s + (0.721 − 0.321i)14-s + (0.176 − 0.542i)15-s + (0.0772 − 0.237i)16-s + (0.687 − 0.306i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(62\)    =    \(2 \cdot 31\)
\( \varepsilon \)  =  $0.659 - 0.751i$
motivic weight  =  \(1\)
character  :  $\chi_{62} (51, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 62,\ (\ :1/2),\ 0.659 - 0.751i)$
$L(1)$  $\approx$  $0.848706 + 0.384291i$
$L(\frac12)$  $\approx$  $0.848706 + 0.384291i$
$L(\frac{3}{2})$   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 \( 1 + (4.84 - 2.74i)T \)
good3 \( 1 + (0.413 + 0.459i)T + (-0.313 + 2.98i)T^{2} \)
5 \( 1 + (-1.78 - 3.09i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.309 + 2.94i)T + (-6.84 + 1.45i)T^{2} \)
11 \( 1 + (2.16 + 0.965i)T + (7.36 + 8.17i)T^{2} \)
13 \( 1 + (5.53 - 1.17i)T + (11.8 - 5.28i)T^{2} \)
17 \( 1 + (-2.83 + 1.26i)T + (11.3 - 12.6i)T^{2} \)
19 \( 1 + (-4.92 - 1.04i)T + (17.3 + 7.72i)T^{2} \)
23 \( 1 + (-1.13 - 0.823i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.07 - 3.29i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + (0.0938 - 0.162i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.60 + 1.78i)T + (-4.28 - 40.7i)T^{2} \)
43 \( 1 + (11.0 + 2.34i)T + (39.2 + 17.4i)T^{2} \)
47 \( 1 + (0.00456 - 0.0140i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.425 - 4.04i)T + (-51.8 - 11.0i)T^{2} \)
59 \( 1 + (-5.00 - 5.56i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 - 13.3T + 61T^{2} \)
67 \( 1 + (1.40 + 2.44i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.911 - 8.67i)T + (-69.4 - 14.7i)T^{2} \)
73 \( 1 + (7.19 + 3.20i)T + (48.8 + 54.2i)T^{2} \)
79 \( 1 + (-6.18 + 2.75i)T + (52.8 - 58.7i)T^{2} \)
83 \( 1 + (-10.3 + 11.4i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (-1.34 + 0.976i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-5.76 + 4.18i)T + (29.9 - 92.2i)T^{2} \)
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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

−14.82260755179035559038236383187, −14.26196728063706115995312733924, −13.30385387526165987366609180335, −11.91725298367593050678223773456, −10.41414720985825017064945705262, −9.614429870941736498595112619909, −7.35275923943755253568160755527, −6.90613850966011603735014023399, −5.44499344304181360403738367522, −3.27308719821372791226150532040, 2.29641160850963261264562225980, 5.05546304975654657425411313163, 5.38473088977579549892485904416, 8.035969737907108567400748932910, 9.440045585932722604894639025987, 10.07375667980508498196595412478, 11.76384356299265510587845974889, 12.64837904613793746237714028512, 13.36768432973709807741916675560, 14.80049455874779289460403092294

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