Properties

Degree 2
Conductor $ 2 \cdot 31 $
Sign $0.190 + 0.981i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

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  :  $\chi_{62} (35, \cdot )$
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 \( 1 + (-7.43 + 5.39i)T + (29.9 - 92.2i)T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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