Properties

Degree 2
Conductor $ 2 \cdot 31 $
Sign $0.190 - 0.981i$
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 + (0.523 + 1.61i)3-s + (−0.809 − 0.587i)4-s − 0.429·5-s − 1.69·6-s + (−0.5 − 0.363i)7-s + (0.809 − 0.587i)8-s + (0.101 − 0.0736i)9-s + (0.132 − 0.408i)10-s + (2.64 + 1.91i)11-s + (0.523 − 1.61i)12-s + (−1.77 − 5.46i)13-s + (0.5 − 0.363i)14-s + (−0.225 − 0.693i)15-s + (0.309 + 0.951i)16-s + (−0.347 + 0.252i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (0.302 + 0.930i)3-s + (−0.404 − 0.293i)4-s − 0.192·5-s − 0.692·6-s + (−0.188 − 0.137i)7-s + (0.286 − 0.207i)8-s + (0.0337 − 0.0245i)9-s + (0.0420 − 0.129i)10-s + (0.796 + 0.578i)11-s + (0.151 − 0.465i)12-s + (−0.492 − 1.51i)13-s + (0.133 − 0.0970i)14-s + (−0.0581 − 0.178i)15-s + (0.0772 + 0.237i)16-s + (−0.0843 + 0.0612i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(62\)    =    \(2 \cdot 31\)
\( \varepsilon \)  =  $0.190 - 0.981i$
motivic weight  =  \(1\)
character  :  $\chi_{62} (39, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 62,\ (\ :1/2),\ 0.190 - 0.981i)$
$L(1)$  $\approx$  $0.639786 + 0.527301i$
$L(\frac12)$  $\approx$  $0.639786 + 0.527301i$
$L(\frac{3}{2})$   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 \( 1 + (-3.64 - 4.21i)T \)
good3 \( 1 + (-0.523 - 1.61i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + 0.429T + 5T^{2} \)
7 \( 1 + (0.5 + 0.363i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (-2.64 - 1.91i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (1.77 + 5.46i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.347 - 0.252i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.48 + 4.55i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (6.68 - 4.85i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.221 + 0.681i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + 7.66T + 37T^{2} \)
41 \( 1 + (1.63 - 5.02i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (1.16 - 3.58i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (1.5 + 4.61i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.82 + 5.68i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.21 + 3.74i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 - 7.62T + 61T^{2} \)
67 \( 1 - 1.79T + 67T^{2} \)
71 \( 1 + (9.12 - 6.63i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-12.5 - 9.11i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.09 + 0.794i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.354 - 1.08i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-5.66 - 4.11i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-7.43 - 5.39i)T + (29.9 + 92.2i)T^{2} \)
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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

−15.44820241103275861633975465497, −14.57273503594673379469427502878, −13.30181117641933176996194540613, −11.88976894034813944548640894160, −10.18921782359690260134789645085, −9.609811292812847741709551595009, −8.229119728852510904416390934452, −6.89723161661783113917978355563, −5.16552983230636990996384603552, −3.71745698967032881466807175778, 1.97597932975552870334453809057, 4.06236322350928226466018868623, 6.39252061699042314790989380678, 7.73617868831625120675354305640, 8.921240537974039880984553265041, 10.17532064075451474424590199337, 11.81633904843037235162703222363, 12.23878936982478228615636038108, 13.73058212685984309782880031456, 14.22849705145379962058449324804

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