↑

Properties

Label 2-31-31.27-c2-0-0
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

Related objects

Downloads

Learn more

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: $\chi_{31} (27, \cdot )$
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 \( 1 + (10.3 - 7.51i)T + (2.90e3 - 8.94e3i)T^{2} \)
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

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

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