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Properties

Label 2-31-31.15-c2-0-4
Degree $2$
Conductor $31$
Sign $0.823 + 0.566i$
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  + (2.95 − 2.14i)2-s + (−2.80 + 3.86i)3-s + (2.87 − 8.85i)4-s − 3.94·5-s + 17.4i·6-s + (−0.932 + 2.87i)7-s + (−5.98 − 18.4i)8-s + (−4.27 − 13.1i)9-s + (−11.6 + 8.45i)10-s + (10.9 + 3.54i)11-s + (26.1 + 35.9i)12-s + (4.53 − 6.23i)13-s + (3.40 + 10.4i)14-s + (11.0 − 15.2i)15-s + (−27.0 − 19.6i)16-s + (−15.6 + 5.07i)17-s + ⋯
L(s)  = 1  + (1.47 − 1.07i)2-s + (−0.936 + 1.28i)3-s + (0.719 − 2.21i)4-s − 0.788·5-s + 2.90i·6-s + (−0.133 + 0.410i)7-s + (−0.748 − 2.30i)8-s + (−0.474 − 1.46i)9-s + (−1.16 + 0.845i)10-s + (0.991 + 0.322i)11-s + (2.17 + 2.99i)12-s + (0.348 − 0.479i)13-s + (0.243 + 0.748i)14-s + (0.738 − 1.01i)15-s + (−1.69 − 1.22i)16-s + (−0.919 + 0.298i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.823 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31\)
Sign: $0.823 + 0.566i$
Analytic conductor: \(0.844688\)
Root analytic conductor: \(0.919069\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 31,\ (\ :1),\ 0.823 + 0.566i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.35806 - 0.422100i\)
\(L(\frac12)\) \(\approx\) \(1.35806 - 0.422100i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 + (-18.2 - 25.0i)T \)
good2 \( 1 + (-2.95 + 2.14i)T + (1.23 - 3.80i)T^{2} \)
3 \( 1 + (2.80 - 3.86i)T + (-2.78 - 8.55i)T^{2} \)
5 \( 1 + 3.94T + 25T^{2} \)
7 \( 1 + (0.932 - 2.87i)T + (-39.6 - 28.8i)T^{2} \)
11 \( 1 + (-10.9 - 3.54i)T + (97.8 + 71.1i)T^{2} \)
13 \( 1 + (-4.53 + 6.23i)T + (-52.2 - 160. i)T^{2} \)
17 \( 1 + (15.6 - 5.07i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-15.7 + 11.4i)T + (111. - 343. i)T^{2} \)
23 \( 1 + (23.9 - 7.77i)T + (427. - 310. i)T^{2} \)
29 \( 1 + (-9.12 - 12.5i)T + (-259. + 799. i)T^{2} \)
37 \( 1 + 48.5iT - 1.36e3T^{2} \)
41 \( 1 + (1.26 - 0.920i)T + (519. - 1.59e3i)T^{2} \)
43 \( 1 + (-33.5 - 46.1i)T + (-571. + 1.75e3i)T^{2} \)
47 \( 1 + (24.4 + 17.7i)T + (682. + 2.10e3i)T^{2} \)
53 \( 1 + (51.0 - 16.5i)T + (2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (54.4 + 39.5i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + 56.2iT - 3.72e3T^{2} \)
67 \( 1 - 87.2T + 4.48e3T^{2} \)
71 \( 1 + (19.9 + 61.2i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (14.3 + 4.67i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (67.5 - 21.9i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (-42.2 - 58.1i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 + (-28.3 - 9.19i)T + (6.40e3 + 4.65e3i)T^{2} \)
97 \( 1 + (-35.5 + 109. i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−15.85406834383283327851893007835, −15.47646294187353217239298061511, −14.16693000024108963030716002684, −12.46515100080643533524126631326, −11.59244144268793183549673663196, −10.86002094548166857315742554628, −9.535752077288405591508338008023, −6.10948838855341509016305248287, −4.72441767353195887927812679403, −3.66966013509481701516777548136, 4.14105537570582838457094359978, 6.02281235007070968721562403662, 6.88567498915409293368706813781, 7.985332976971839303835356808636, 11.62407200475651824683455437808, 11.98894659252871578768783374902, 13.35789901553840194369778593023, 14.08474720259771948467720332691, 15.67259393930370316972690547898, 16.60993005198089145090575025478

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