↑

Properties

Label 2-31e2-31.4-c1-0-23
Degree $2$
Conductor $961$
Sign $0.501 - 0.865i$
Analytic cond. $7.67362$
Root an. cond. $2.77013$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.391 + 1.20i)2-s + (0.458 − 1.41i)3-s + (0.320 − 0.232i)4-s − 3.80·5-s + 1.88·6-s + (−1.77 + 1.28i)7-s + (2.45 + 1.78i)8-s + (0.644 + 0.468i)9-s + (−1.48 − 4.57i)10-s + (0.769 − 0.558i)11-s + (−0.181 − 0.558i)12-s + (0.0519 − 0.159i)13-s + (−2.24 − 1.62i)14-s + (−1.74 + 5.36i)15-s + (−0.943 + 2.90i)16-s + (5.32 + 3.86i)17-s + ⋯
L(s)  = 1  + (0.276 + 0.851i)2-s + (0.264 − 0.815i)3-s + (0.160 − 0.116i)4-s − 1.69·5-s + 0.767·6-s + (−0.669 + 0.486i)7-s + (0.867 + 0.630i)8-s + (0.214 + 0.156i)9-s + (−0.470 − 1.44i)10-s + (0.231 − 0.168i)11-s + (−0.0523 − 0.161i)12-s + (0.0144 − 0.0443i)13-s + (−0.599 − 0.435i)14-s + (−0.450 + 1.38i)15-s + (−0.235 + 0.725i)16-s + (1.29 + 0.937i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.501 - 0.865i$
Analytic conductor: \(7.67362\)
Root analytic conductor: \(2.77013\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :1/2),\ 0.501 - 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46860 + 0.846423i\)
\(L(\frac12)\) \(\approx\) \(1.46860 + 0.846423i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + (-0.391 - 1.20i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.458 + 1.41i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + 3.80T + 5T^{2} \)
7 \( 1 + (1.77 - 1.28i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-0.769 + 0.558i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.0519 + 0.159i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-5.32 - 3.86i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.356 + 1.09i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-3.74 - 2.72i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.413 - 1.27i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + 3.87T + 37T^{2} \)
41 \( 1 + (-0.101 - 0.312i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-2.97 - 9.15i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-1.74 + 5.36i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.93 - 4.31i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.819 - 2.52i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + 1.74T + 61T^{2} \)
67 \( 1 - 0.552T + 67T^{2} \)
71 \( 1 + (0.919 + 0.668i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-6.40 + 4.65i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.67 + 2.66i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.100 - 0.310i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-11.9 + 8.64i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (12.5 - 9.12i)T + (29.9 - 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.28283359697501905323184405398, −8.929136133135624405578863712825, −8.043274896546945712986944866052, −7.59795075587895492911180740981, −6.92113837144512785180013713152, −6.12511618370272725087871101062, −5.05653892291422283628166090283, −3.93950194209180992917566330743, −2.89582085733345213315121162548, −1.27749348790612967915597221255, 0.835431649903740984266659843376, 2.89688837546857195956555442661, 3.63802890890178464906632472779, 4.04322115518178739833087516718, 4.99481261369495254096339867018, 6.86845495317616279333615345225, 7.27753758383668985290126306329, 8.224647989473540982031524574620, 9.305778002964025298047167691104, 10.11720112452119281228754758209

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