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Properties

Label 2-31-31.23-c2-0-0
Degree $2$
Conductor $31$
Sign $-0.770 - 0.637i$
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  + (−0.833 + 2.56i)2-s + (−3.19 + 1.03i)3-s + (−2.64 − 1.92i)4-s − 1.06·5-s − 9.06i·6-s + (10.2 + 7.47i)7-s + (−1.58 + 1.15i)8-s + (1.85 − 1.34i)9-s + (0.887 − 2.73i)10-s + (3.94 − 5.42i)11-s + (10.4 + 3.39i)12-s + (13.8 − 4.49i)13-s + (−27.7 + 20.1i)14-s + (3.40 − 1.10i)15-s + (−5.67 − 17.4i)16-s + (−8.21 − 11.3i)17-s + ⋯
L(s)  = 1  + (−0.416 + 1.28i)2-s + (−1.06 + 0.346i)3-s + (−0.661 − 0.480i)4-s − 0.212·5-s − 1.51i·6-s + (1.46 + 1.06i)7-s + (−0.198 + 0.144i)8-s + (0.205 − 0.149i)9-s + (0.0887 − 0.273i)10-s + (0.358 − 0.493i)11-s + (0.871 + 0.283i)12-s + (1.06 − 0.345i)13-s + (−1.98 + 1.44i)14-s + (0.226 − 0.0737i)15-s + (−0.354 − 1.09i)16-s + (−0.483 − 0.665i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31\)
Sign: $-0.770 - 0.637i$
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: \((2,\ 31,\ (\ :1),\ -0.770 - 0.637i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.218544 + 0.606591i\)
\(L(\frac12)\) \(\approx\) \(0.218544 + 0.606591i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 + (-29.5 + 9.26i)T \)
good2 \( 1 + (0.833 - 2.56i)T + (-3.23 - 2.35i)T^{2} \)
3 \( 1 + (3.19 - 1.03i)T + (7.28 - 5.29i)T^{2} \)
5 \( 1 + 1.06T + 25T^{2} \)
7 \( 1 + (-10.2 - 7.47i)T + (15.1 + 46.6i)T^{2} \)
11 \( 1 + (-3.94 + 5.42i)T + (-37.3 - 115. i)T^{2} \)
13 \( 1 + (-13.8 + 4.49i)T + (136. - 99.3i)T^{2} \)
17 \( 1 + (8.21 + 11.3i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (1.96 - 6.03i)T + (-292. - 212. i)T^{2} \)
23 \( 1 + (-23.0 - 31.7i)T + (-163. + 503. i)T^{2} \)
29 \( 1 + (4.99 + 1.62i)T + (680. + 494. i)T^{2} \)
37 \( 1 + 17.1iT - 1.36e3T^{2} \)
41 \( 1 + (-12.7 + 39.2i)T + (-1.35e3 - 988. i)T^{2} \)
43 \( 1 + (43.4 + 14.1i)T + (1.49e3 + 1.08e3i)T^{2} \)
47 \( 1 + (-14.6 - 45.0i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (2.71 + 3.73i)T + (-868. + 2.67e3i)T^{2} \)
59 \( 1 + (27.9 + 86.0i)T + (-2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + 27.7iT - 3.72e3T^{2} \)
67 \( 1 - 1.15T + 4.48e3T^{2} \)
71 \( 1 + (27.4 - 19.9i)T + (1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (-9.88 + 13.6i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (39.0 + 53.8i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (-20.4 - 6.64i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 + (-33.2 + 45.7i)T + (-2.44e3 - 7.53e3i)T^{2} \)
97 \( 1 + (26.0 + 18.9i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−17.25481614856261422187604587737, −15.89615284100373592784597896484, −15.35456182770425093821009921977, −14.00584851363693796605414701082, −11.72319781984570021553199291101, −11.20796261078861581191878642872, −8.967067542722350772391840428947, −7.906010467461407410219266344692, −6.05128559128015105208426338881, −5.16147170040855440083155358833, 1.25773917290059582817309957825, 4.36249436859128197424038151851, 6.62332685665442840821010011721, 8.535508366716424134611489566333, 10.49905705514580255652701311957, 11.20092922345363437801842702417, 11.93145192433760984116250198858, 13.34187182615296496445606526061, 14.95290936197938622542918232255, 16.84722312957476723161218995092

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