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Properties

Label 2-31-31.26-c2-0-1
Degree $2$
Conductor $31$
Sign $0.972 - 0.232i$
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  + 0.571·2-s + (2.95 + 1.70i)3-s − 3.67·4-s + (−0.928 − 1.60i)5-s + (1.69 + 0.977i)6-s + (1.95 − 3.39i)7-s − 4.38·8-s + (1.33 + 2.31i)9-s + (−0.530 − 0.919i)10-s + (−1.86 + 1.07i)11-s + (−10.8 − 6.27i)12-s + (−14.8 + 8.55i)13-s + (1.12 − 1.94i)14-s − 6.34i·15-s + 12.1·16-s + (18.6 + 10.7i)17-s + ⋯
L(s)  = 1  + 0.285·2-s + (0.986 + 0.569i)3-s − 0.918·4-s + (−0.185 − 0.321i)5-s + (0.282 + 0.162i)6-s + (0.279 − 0.484i)7-s − 0.548·8-s + (0.148 + 0.257i)9-s + (−0.0530 − 0.0919i)10-s + (−0.169 + 0.0980i)11-s + (−0.905 − 0.522i)12-s + (−1.13 + 0.657i)13-s + (0.0800 − 0.138i)14-s − 0.422i·15-s + 0.761·16-s + (1.09 + 0.634i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.232i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.972 - 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31\)
Sign: $0.972 - 0.232i$
Analytic conductor: \(0.844688\)
Root analytic conductor: \(0.919069\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 31,\ (\ :1),\ 0.972 - 0.232i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.17972 + 0.138957i\)
\(L(\frac12)\) \(\approx\) \(1.17972 + 0.138957i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 + (26.1 + 16.6i)T \)
good2 \( 1 - 0.571T + 4T^{2} \)
3 \( 1 + (-2.95 - 1.70i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (0.928 + 1.60i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-1.95 + 3.39i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (1.86 - 1.07i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (14.8 - 8.55i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-18.6 - 10.7i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-9.77 + 16.9i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 - 43.6iT - 529T^{2} \)
29 \( 1 - 14.6iT - 841T^{2} \)
37 \( 1 + (-4.67 - 2.69i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (1.07 + 1.85i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (50.3 + 29.0i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + 48.7T + 2.20e3T^{2} \)
53 \( 1 + (-61.9 + 35.7i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (11.8 - 20.4i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + 31.6iT - 3.72e3T^{2} \)
67 \( 1 + (2.58 + 4.47i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (50.5 + 87.4i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-2.75 + 1.59i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-14.4 - 8.31i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-46.4 + 26.7i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 174. iT - 7.92e3T^{2} \)
97 \( 1 - 135.T + 9.40e3T^{2} \)
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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

−16.72141946546241721426993547557, −15.14221016466636229078682743118, −14.36737509736926047330799090552, −13.41526847189358162811723797375, −11.99637709284407264881122424490, −9.946296441173999956655502742247, −9.065766159421978606503133484250, −7.72803847441625650992145326645, −5.00298742885597029790160593204, −3.61192892030772775712027241207, 3.02098598332764219913449069511, 5.23779550101302809633866492861, 7.58579595937080277018591390007, 8.611616850907729113339927069651, 10.03105679270013650511045223592, 12.13178693575231859628951562943, 13.10956875573515969273714832423, 14.49258820476991308994282944450, 14.69438035794120346457451301372, 16.70683308967812736295322777410

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