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Properties

Label 2-31-31.23-c2-0-3
Degree $2$
Conductor $31$
Sign $0.596 + 0.802i$
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  + (0.592 − 1.82i)2-s + (1.04 − 0.339i)3-s + (0.259 + 0.188i)4-s − 6.67·5-s − 2.10i·6-s + (3.63 + 2.63i)7-s + (6.70 − 4.87i)8-s + (−6.30 + 4.58i)9-s + (−3.95 + 12.1i)10-s + (−5.88 + 8.10i)11-s + (0.335 + 0.108i)12-s + (8.30 − 2.69i)13-s + (6.96 − 5.05i)14-s + (−6.97 + 2.26i)15-s + (−4.51 − 13.8i)16-s + (−6.86 − 9.45i)17-s + ⋯
L(s)  = 1  + (0.296 − 0.912i)2-s + (0.348 − 0.113i)3-s + (0.0648 + 0.0471i)4-s − 1.33·5-s − 0.350i·6-s + (0.518 + 0.376i)7-s + (0.838 − 0.608i)8-s + (−0.700 + 0.509i)9-s + (−0.395 + 1.21i)10-s + (−0.535 + 0.736i)11-s + (0.0279 + 0.00907i)12-s + (0.639 − 0.207i)13-s + (0.497 − 0.361i)14-s + (−0.464 + 0.151i)15-s + (−0.282 − 0.868i)16-s + (−0.404 − 0.556i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31\)
Sign: $0.596 + 0.802i$
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: \((2,\ 31,\ (\ :1),\ 0.596 + 0.802i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.03308 - 0.519239i\)
\(L(\frac12)\) \(\approx\) \(1.03308 - 0.519239i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 + (30.9 - 1.96i)T \)
good2 \( 1 + (-0.592 + 1.82i)T + (-3.23 - 2.35i)T^{2} \)
3 \( 1 + (-1.04 + 0.339i)T + (7.28 - 5.29i)T^{2} \)
5 \( 1 + 6.67T + 25T^{2} \)
7 \( 1 + (-3.63 - 2.63i)T + (15.1 + 46.6i)T^{2} \)
11 \( 1 + (5.88 - 8.10i)T + (-37.3 - 115. i)T^{2} \)
13 \( 1 + (-8.30 + 2.69i)T + (136. - 99.3i)T^{2} \)
17 \( 1 + (6.86 + 9.45i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (-5.34 + 16.4i)T + (-292. - 212. i)T^{2} \)
23 \( 1 + (16.6 + 22.9i)T + (-163. + 503. i)T^{2} \)
29 \( 1 + (-46.5 - 15.1i)T + (680. + 494. i)T^{2} \)
37 \( 1 - 51.5iT - 1.36e3T^{2} \)
41 \( 1 + (-13.2 + 40.8i)T + (-1.35e3 - 988. i)T^{2} \)
43 \( 1 + (-33.9 - 11.0i)T + (1.49e3 + 1.08e3i)T^{2} \)
47 \( 1 + (-16.6 - 51.1i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-44.3 - 61.0i)T + (-868. + 2.67e3i)T^{2} \)
59 \( 1 + (22.2 + 68.6i)T + (-2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 - 47.1iT - 3.72e3T^{2} \)
67 \( 1 + 61.3T + 4.48e3T^{2} \)
71 \( 1 + (77.7 - 56.5i)T + (1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (-28.2 + 38.8i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (-16.9 - 23.3i)T + (-1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (28.7 + 9.34i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 + (23.4 - 32.3i)T + (-2.44e3 - 7.53e3i)T^{2} \)
97 \( 1 + (85.4 + 62.1i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−16.22606025281041246920851479868, −15.38490949049045265348882714407, −13.89703408399955523818989235426, −12.51901397988632325671631334405, −11.58017192244969740039204692946, −10.69418474835318749739133655643, −8.506864873961537330569207032146, −7.39418448200317129390851820418, −4.56409813501147942182695070919, −2.75568519170733192107918360870, 3.92461622529077347642715063194, 5.89879143522899353106696649111, 7.64361003964204977033130541277, 8.440818087640020182573689360946, 10.81548878347889292303113422206, 11.77705425528929823444061462389, 13.74177806142023066233268489737, 14.67595519628529368155507418153, 15.67057303767546207341167857517, 16.38152576862499312880454717740

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