Properties

Degree 2
Conductor $ 2 \cdot 31 $
Sign $0.747 - 0.664i$
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 + (0.604 − 0.128i)3-s + (−0.809 + 0.587i)4-s + (0.139 − 0.242i)5-s + (0.309 + 0.535i)6-s + (−0.309 − 0.137i)7-s + (−0.809 − 0.587i)8-s + (−2.39 + 1.06i)9-s + (0.273 + 0.0581i)10-s + (−0.521 − 4.96i)11-s + (−0.413 + 0.459i)12-s + (1.05 + 1.17i)13-s + (0.0353 − 0.336i)14-s + (0.0534 − 0.164i)15-s + (0.309 − 0.951i)16-s + (0.0201 − 0.191i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.349 − 0.0741i)3-s + (−0.404 + 0.293i)4-s + (0.0625 − 0.108i)5-s + (0.126 + 0.218i)6-s + (−0.116 − 0.0520i)7-s + (−0.286 − 0.207i)8-s + (−0.797 + 0.354i)9-s + (0.0865 + 0.0183i)10-s + (−0.157 − 1.49i)11-s + (−0.119 + 0.132i)12-s + (0.293 + 0.326i)13-s + (0.00944 − 0.0899i)14-s + (0.0137 − 0.0424i)15-s + (0.0772 − 0.237i)16-s + (0.00487 − 0.0463i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(62\)    =    \(2 \cdot 31\)
\( \varepsilon \)  =  $0.747 - 0.664i$
motivic weight  =  \(1\)
character  :  $\chi_{62} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 62,\ (\ :1/2),\ 0.747 - 0.664i)$
$L(1)$  $\approx$  $0.924937 + 0.351935i$
$L(\frac12)$  $\approx$  $0.924937 + 0.351935i$
$L(\frac{3}{2})$   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 \( 1 + (-4.79 - 2.82i)T \)
good3 \( 1 + (-0.604 + 0.128i)T + (2.74 - 1.22i)T^{2} \)
5 \( 1 + (-0.139 + 0.242i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.309 + 0.137i)T + (4.68 + 5.20i)T^{2} \)
11 \( 1 + (0.521 + 4.96i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (-1.05 - 1.17i)T + (-1.35 + 12.9i)T^{2} \)
17 \( 1 + (-0.0201 + 0.191i)T + (-16.6 - 3.53i)T^{2} \)
19 \( 1 + (0.644 - 0.716i)T + (-1.98 - 18.8i)T^{2} \)
23 \( 1 + (3.17 + 2.30i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.23 - 6.88i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + (-3.82 - 6.63i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (8.03 + 1.70i)T + (37.4 + 16.6i)T^{2} \)
43 \( 1 + (-2.29 + 2.54i)T + (-4.49 - 42.7i)T^{2} \)
47 \( 1 + (-3.04 + 9.38i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-8.58 + 3.82i)T + (35.4 - 39.3i)T^{2} \)
59 \( 1 + (10.8 - 2.30i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + 1.36T + 61T^{2} \)
67 \( 1 + (-3.29 + 5.70i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.22 - 2.32i)T + (47.5 - 52.7i)T^{2} \)
73 \( 1 + (1.61 + 15.4i)T + (-71.4 + 15.1i)T^{2} \)
79 \( 1 + (1.42 - 13.5i)T + (-77.2 - 16.4i)T^{2} \)
83 \( 1 + (-14.2 - 3.02i)T + (75.8 + 33.7i)T^{2} \)
89 \( 1 + (-4.00 + 2.91i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (9.83 - 7.14i)T + (29.9 - 92.2i)T^{2} \)
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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

−15.06561825708223632831663632690, −13.95187743330107512844517594783, −13.43330040134656267641092996074, −11.89711038422715079734856153691, −10.62083847222076835516249752918, −8.864208919682163019070498697581, −8.200309469133326217653128466250, −6.54468595342712254870122971035, −5.27991098441012014176084387204, −3.29481454569935208688558704031, 2.59568837580838752382092854010, 4.35427544419291766959026278234, 6.10544036322433664435624939630, 7.928790537078734307059368518054, 9.326241864569601123174954715538, 10.28025178774725118363832058176, 11.65108925959341122525945478337, 12.58660107069697330193968389618, 13.76207559497010012145013863324, 14.76438561797044982500561441123

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