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Properties

Label 1-31-31.26-r1-0-0
Degree $1$
Conductor $31$
Sign $0.695 + 0.718i$
Analytic cond. $3.33141$
Root an. cond. $3.33141$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)12-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s − 15-s + 16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + 2-s + (0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)12-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s − 15-s + 16-s + (0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(31\)
Sign: $0.695 + 0.718i$
Analytic conductor: \(3.33141\)
Root analytic conductor: \(3.33141\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{31} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 31,\ (1:\ ),\ 0.695 + 0.718i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.371469482 + 1.005786976i\)
\(L(\frac12)\) \(\approx\) \(2.371469482 + 1.005786976i\)
\(L(1)\) \(\approx\) \(1.879213610 + 0.5376320614i\)
\(L(1)\) \(\approx\) \(1.879213610 + 0.5376320614i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 - T \)
29 \( 1 - T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 - T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−36.03694199047081809395459724281, −35.30864266593074978771591649753, −33.8241549331911259626327357322, −32.129778633924270460121683213657, −31.54489954313244131089796391377, −30.53263976307618275070690829051, −29.16141499249758529834513709123, −28.0945904790797707352893286350, −25.69070979248761182016100995745, −24.904979113067120430837549644393, −23.806582542939423420906081340444, −22.72252967739608720611109521201, −20.96390856165423950362814455136, −19.94453395593242918465398300995, −18.735126484161719053692078780397, −16.60173698608234703190103783223, −15.25089922348459432110354945030, −13.879620718504668161994951672300, −12.44207778956765682717281271220, −11.949121923483919535504377485389, −9.11916197987811066740563336970, −7.45087480449111406298688286638, −5.89688076625617776774994700343, −3.90954114053089609500609102833, −1.93601721019922475920401420063, 3.14900146449111918990716082501, 4.02909061054478038079593727430, 6.16852612519315094017700013557, 7.90843428682917089599907565647, 10.30301459129209214350523835223, 11.220089817316603931959723291699, 13.32190208552506637014404100721, 14.4786587114416088206168908723, 15.53818190195904911978227263916, 16.717895780434211211143117394807, 19.33388309535322849716732648173, 20.20684450635459460249475492654, 21.71749720389530663485905264589, 22.57729055762610550776387144847, 23.77646257686247141114177455735, 25.59696995015023513032903862194, 26.44017494393920563813497510448, 27.88533073949353888140756660178, 29.83973207742039441096728534060, 30.51459276150292375835453676759, 32.05994899404862366440308541101, 32.667001701834777095473969683222, 33.91330850741717403494440774979, 35.06770270199252064084748377345, 37.1672086619915527437976882128

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