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Properties

Label 2-31-31.23-c2-0-4
Degree $2$
Conductor $31$
Sign $-0.234 + 0.972i$
Analytic cond. $0.844688$
Root an. cond. $0.919069$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.09 − 3.36i)2-s + (−3.38 + 1.10i)3-s + (−6.88 − 5.00i)4-s + 8.82·5-s + 12.5i·6-s + (2.08 + 1.51i)7-s + (−12.9 + 9.37i)8-s + (2.98 − 2.16i)9-s + (9.64 − 29.6i)10-s + (−6.28 + 8.64i)11-s + (28.8 + 9.36i)12-s + (−2.12 + 0.689i)13-s + (7.36 − 5.34i)14-s + (−29.9 + 9.71i)15-s + (6.91 + 21.2i)16-s + (−0.840 − 1.15i)17-s + ⋯
L(s)  = 1  + (0.546 − 1.68i)2-s + (−1.12 + 0.366i)3-s + (−1.72 − 1.25i)4-s + 1.76·5-s + 2.09i·6-s + (0.297 + 0.216i)7-s + (−1.61 + 1.17i)8-s + (0.331 − 0.240i)9-s + (0.964 − 2.96i)10-s + (−0.571 + 0.786i)11-s + (2.40 + 0.780i)12-s + (−0.163 + 0.0530i)13-s + (0.525 − 0.382i)14-s + (−1.99 + 0.647i)15-s + (0.432 + 1.33i)16-s + (−0.0494 − 0.0680i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.234 + 0.972i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.234 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31\)
Sign: $-0.234 + 0.972i$
Analytic conductor: \(0.844688\)
Root analytic conductor: \(0.919069\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 31,\ (\ :1),\ -0.234 + 0.972i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.650208 - 0.826108i\)
\(L(\frac12)\) \(\approx\) \(0.650208 - 0.826108i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 + (21.3 + 22.5i)T \)
good2 \( 1 + (-1.09 + 3.36i)T + (-3.23 - 2.35i)T^{2} \)
3 \( 1 + (3.38 - 1.10i)T + (7.28 - 5.29i)T^{2} \)
5 \( 1 - 8.82T + 25T^{2} \)
7 \( 1 + (-2.08 - 1.51i)T + (15.1 + 46.6i)T^{2} \)
11 \( 1 + (6.28 - 8.64i)T + (-37.3 - 115. i)T^{2} \)
13 \( 1 + (2.12 - 0.689i)T + (136. - 99.3i)T^{2} \)
17 \( 1 + (0.840 + 1.15i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (2.52 - 7.77i)T + (-292. - 212. i)T^{2} \)
23 \( 1 + (1.14 + 1.58i)T + (-163. + 503. i)T^{2} \)
29 \( 1 + (26.7 + 8.69i)T + (680. + 494. i)T^{2} \)
37 \( 1 + 6.07iT - 1.36e3T^{2} \)
41 \( 1 + (20.7 - 63.7i)T + (-1.35e3 - 988. i)T^{2} \)
43 \( 1 + (-55.9 - 18.1i)T + (1.49e3 + 1.08e3i)T^{2} \)
47 \( 1 + (11.9 + 36.6i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-45.0 - 61.9i)T + (-868. + 2.67e3i)T^{2} \)
59 \( 1 + (7.78 + 23.9i)T + (-2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + 87.4iT - 3.72e3T^{2} \)
67 \( 1 - 31.7T + 4.48e3T^{2} \)
71 \( 1 + (51.7 - 37.6i)T + (1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (-42.4 + 58.4i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (54.8 + 75.4i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (-85.7 - 27.8i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 + (35.3 - 48.6i)T + (-2.44e3 - 7.53e3i)T^{2} \)
97 \( 1 + (10.3 + 7.51i)T + (2.90e3 + 8.94e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.80538029451318032231257899848, −14.66758019119287986839306167026, −13.44251985091496988624643120389, −12.52450944956056778134217446081, −11.24939067058898697624381779240, −10.28140880613897735672758288818, −9.516389633410564606289501643166, −5.82601760225289055298011519958, −4.86579026998417329568042448955, −2.12034115786327672839296047268, 5.28475679273230338214976484058, 5.88160340986733699026772125990, 7.05470446535861459373271899081, 8.925639012042783820781421285370, 10.73877619387229532185891099411, 12.74160816174521247344831905972, 13.62034194921819939345697477693, 14.53472967887972232498957298495, 16.14549361257150976050857137421, 17.08721455270530225694850289406

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