Properties

Label 1-968-968.515-r1-0-0
Degree $1$
Conductor $968$
Sign $0.711 - 0.702i$
Analytic cond. $104.026$
Root an. cond. $104.026$
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  + (0.309 − 0.951i)3-s + (−0.610 + 0.791i)5-s + (0.985 − 0.170i)7-s + (−0.809 − 0.587i)9-s + (−0.974 + 0.226i)13-s + (0.564 + 0.825i)15-s + (−0.362 + 0.931i)17-s + (0.774 − 0.633i)19-s + (0.142 − 0.989i)21-s + (0.142 + 0.989i)23-s + (−0.254 − 0.967i)25-s + (−0.809 + 0.587i)27-s + (0.254 − 0.967i)29-s + (−0.0855 − 0.996i)31-s + (−0.466 + 0.884i)35-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)3-s + (−0.610 + 0.791i)5-s + (0.985 − 0.170i)7-s + (−0.809 − 0.587i)9-s + (−0.974 + 0.226i)13-s + (0.564 + 0.825i)15-s + (−0.362 + 0.931i)17-s + (0.774 − 0.633i)19-s + (0.142 − 0.989i)21-s + (0.142 + 0.989i)23-s + (−0.254 − 0.967i)25-s + (−0.809 + 0.587i)27-s + (0.254 − 0.967i)29-s + (−0.0855 − 0.996i)31-s + (−0.466 + 0.884i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.711 - 0.702i$
Analytic conductor: \(104.026\)
Root analytic conductor: \(104.026\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 968,\ (1:\ ),\ 0.711 - 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.846393034 - 0.7577899379i\)
\(L(\frac12)\) \(\approx\) \(1.846393034 - 0.7577899379i\)
\(L(1)\) \(\approx\) \(1.095656194 - 0.2278661151i\)
\(L(1)\) \(\approx\) \(1.095656194 - 0.2278661151i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (-0.610 + 0.791i)T \)
7 \( 1 + (0.985 - 0.170i)T \)
13 \( 1 + (-0.974 + 0.226i)T \)
17 \( 1 + (-0.362 + 0.931i)T \)
19 \( 1 + (0.774 - 0.633i)T \)
23 \( 1 + (0.142 + 0.989i)T \)
29 \( 1 + (0.254 - 0.967i)T \)
31 \( 1 + (-0.0855 - 0.996i)T \)
37 \( 1 + (-0.516 + 0.856i)T \)
41 \( 1 + (0.993 - 0.113i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + (0.736 + 0.676i)T \)
53 \( 1 + (-0.696 + 0.717i)T \)
59 \( 1 + (0.993 + 0.113i)T \)
61 \( 1 + (-0.198 - 0.980i)T \)
67 \( 1 + (0.415 + 0.909i)T \)
71 \( 1 + (0.998 - 0.0570i)T \)
73 \( 1 + (0.897 - 0.441i)T \)
79 \( 1 + (-0.941 - 0.336i)T \)
83 \( 1 + (-0.466 - 0.884i)T \)
89 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (0.610 + 0.791i)T \)
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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

−21.441303151556244917989030774691, −20.858890660480559041883417935258, −20.10620418587906793048014627545, −19.71377771096434307258863922967, −18.46868690166371190649329750508, −17.61151249291987226848891878775, −16.70036819433638105384743516232, −16.1134287697921747938332559936, −15.382793347129305885823011262885, −14.48820883101628121130945311494, −14.053851331141357511981285006781, −12.69325904111520866120023786451, −11.94603917944419447659802840452, −11.203973050159285097692573747716, −10.32820742707032835455431911713, −9.35856292695754408161638632519, −8.6604222852088843921744671137, −7.96562301081736127406292119494, −7.09352601981152625735628804482, −5.31803028113561141040473680255, −5.040432413117147279270629058086, −4.207950321820229148048525860383, −3.17044201828534543914218405176, −2.09028768083743031996521096330, −0.70353426587747898827173368768, 0.59360333641261126791930649478, 1.82648152754279206711776964841, 2.61828266032561415638382574083, 3.67256943646116158332336267234, 4.70585614818202544581646950178, 5.886953182192806048722541477173, 6.86936019931101809304553681620, 7.59932605665453705992340140440, 8.0312281857722020900591765114, 9.117994728708041800374918273421, 10.207421273301466253464251494377, 11.43118490694199423366505802864, 11.56412409257262594972163507035, 12.624195170736438271423655739144, 13.63975353852816322312556990630, 14.25648079992648723304453316285, 15.021563044357997245769781702817, 15.58624034813883878432733564910, 17.244853929442895590578719625722, 17.420443777796830655746351459342, 18.44277765908094427828715188466, 19.08913461030475304070485092422, 19.7592445228865288727875068920, 20.4232137561278958309518407050, 21.52747511668332027452728322745

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