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Properties

Label 1-31e2-961.164-r0-0-0
Degree $1$
Conductor $961$
Sign $-0.871 + 0.490i$
Analytic cond. $4.46286$
Root an. cond. $4.46286$
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  + (0.992 + 0.121i)2-s + (−0.880 + 0.473i)3-s + (0.970 + 0.240i)4-s + (−0.972 + 0.234i)5-s + (−0.931 + 0.363i)6-s + (−0.590 + 0.807i)7-s + (0.934 + 0.356i)8-s + (0.551 − 0.834i)9-s + (−0.993 + 0.114i)10-s + (0.606 + 0.794i)11-s + (−0.968 + 0.247i)12-s + (0.988 + 0.148i)13-s + (−0.684 + 0.729i)14-s + (0.745 − 0.666i)15-s + (0.884 + 0.467i)16-s + (−0.916 + 0.400i)17-s + ⋯
L(s)  = 1  + (0.992 + 0.121i)2-s + (−0.880 + 0.473i)3-s + (0.970 + 0.240i)4-s + (−0.972 + 0.234i)5-s + (−0.931 + 0.363i)6-s + (−0.590 + 0.807i)7-s + (0.934 + 0.356i)8-s + (0.551 − 0.834i)9-s + (−0.993 + 0.114i)10-s + (0.606 + 0.794i)11-s + (−0.968 + 0.247i)12-s + (0.988 + 0.148i)13-s + (−0.684 + 0.729i)14-s + (0.745 − 0.666i)15-s + (0.884 + 0.467i)16-s + (−0.916 + 0.400i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $-0.871 + 0.490i$
Analytic conductor: \(4.46286\)
Root analytic conductor: \(4.46286\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 961,\ (0:\ ),\ -0.871 + 0.490i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3601423734 + 1.373665315i\)
\(L(\frac12)\) \(\approx\) \(0.3601423734 + 1.373665315i\)
\(L(1)\) \(\approx\) \(1.029418118 + 0.6328260643i\)
\(L(1)\) \(\approx\) \(1.029418118 + 0.6328260643i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + (0.992 + 0.121i)T \)
3 \( 1 + (-0.880 + 0.473i)T \)
5 \( 1 + (-0.972 + 0.234i)T \)
7 \( 1 + (-0.590 + 0.807i)T \)
11 \( 1 + (0.606 + 0.794i)T \)
13 \( 1 + (0.988 + 0.148i)T \)
17 \( 1 + (-0.916 + 0.400i)T \)
19 \( 1 + (0.296 + 0.955i)T \)
23 \( 1 + (-0.476 - 0.879i)T \)
29 \( 1 + (-0.983 + 0.181i)T \)
37 \( 1 + (0.963 - 0.266i)T \)
41 \( 1 + (0.359 + 0.932i)T \)
43 \( 1 + (-0.197 - 0.980i)T \)
47 \( 1 + (-0.131 + 0.991i)T \)
53 \( 1 + (-0.997 + 0.0742i)T \)
59 \( 1 + (-0.622 + 0.782i)T \)
61 \( 1 + (0.347 - 0.937i)T \)
67 \( 1 + (-0.315 + 0.948i)T \)
71 \( 1 + (0.321 + 0.946i)T \)
73 \( 1 + (-0.921 + 0.388i)T \)
79 \( 1 + (-0.921 - 0.388i)T \)
83 \( 1 + (-0.601 + 0.798i)T \)
89 \( 1 + (-0.211 - 0.977i)T \)
97 \( 1 + (-0.643 + 0.765i)T \)
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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

−21.77803923438010428975719014454, −20.60912312809924763203341344272, −19.80864491240070563328354555076, −19.39176249758716061373416431108, −18.42119044990399723763018303843, −17.27221665145541215677637851906, −16.41091696132774702528827119411, −15.99458081653608083203587111616, −15.26753835953160179705488939322, −13.88664859297691246647577301643, −13.34949340434299115919927305006, −12.77119122325543118008555144109, −11.61366671795744191623175044629, −11.31769366008698601203409516861, −10.67874288645191237540710170447, −9.33768090742024254081419709727, −7.973679198148861632550260253192, −7.170367615515752956178211760317, −6.49585156645390360027284693383, −5.71345171583632057094063642659, −4.60210176963621572140775100652, −3.916139072822478363442323645312, −3.08359660490432283738100452140, −1.49828535977673375429006769499, −0.50900584032921864558386510873, 1.57802784264142397593736195194, 2.94623618678013380977112257143, 4.03021398562961790108184144748, 4.284487074905639554578384286011, 5.5751243498336329265501268116, 6.3115469184530324239325316055, 6.87725903162436396840453133873, 8.04148061884023619323606427384, 9.162238746559899049884387951360, 10.275978872165047491039644724, 11.17554764531520419050180116132, 11.697111485567716448123039053034, 12.49364648537505413112186825516, 12.955477317962469574051838047723, 14.47099088602127676878596506783, 15.01115150590326787088661069890, 15.85118804345519840844850178383, 16.15150669616649922117487575812, 17.06386073331731115785102099746, 18.21887801738279101426013997483, 18.92667680336431164340288452746, 20.08265234558637149785520058195, 20.55786325911536941857445372961, 21.670099736464014611737854676563, 22.30155001475215956832944568330

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