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Properties

Label 1-941-941.868-r0-0-0
Degree $1$
Conductor $941$
Sign $0.452 + 0.891i$
Analytic cond. $4.36998$
Root an. cond. $4.36998$
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  + (0.964 + 0.264i)2-s + (0.359 − 0.933i)3-s + (0.860 + 0.509i)4-s + (0.480 + 0.876i)5-s + (0.593 − 0.805i)6-s + (−0.997 + 0.0667i)7-s + (0.695 + 0.718i)8-s + (−0.741 − 0.670i)9-s + (0.231 + 0.972i)10-s + (0.593 + 0.805i)11-s + (0.784 − 0.619i)12-s + (−0.892 + 0.451i)13-s + (−0.979 − 0.199i)14-s + (0.991 − 0.133i)15-s + (0.480 + 0.876i)16-s + (−0.979 − 0.199i)17-s + ⋯
L(s)  = 1  + (0.964 + 0.264i)2-s + (0.359 − 0.933i)3-s + (0.860 + 0.509i)4-s + (0.480 + 0.876i)5-s + (0.593 − 0.805i)6-s + (−0.997 + 0.0667i)7-s + (0.695 + 0.718i)8-s + (−0.741 − 0.670i)9-s + (0.231 + 0.972i)10-s + (0.593 + 0.805i)11-s + (0.784 − 0.619i)12-s + (−0.892 + 0.451i)13-s + (−0.979 − 0.199i)14-s + (0.991 − 0.133i)15-s + (0.480 + 0.876i)16-s + (−0.979 − 0.199i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 941 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 941 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(941\)
Sign: $0.452 + 0.891i$
Analytic conductor: \(4.36998\)
Root analytic conductor: \(4.36998\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{941} (868, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 941,\ (0:\ ),\ 0.452 + 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.421913864 + 1.487072526i\)
\(L(\frac12)\) \(\approx\) \(2.421913864 + 1.487072526i\)
\(L(1)\) \(\approx\) \(1.928961029 + 0.4387365454i\)
\(L(1)\) \(\approx\) \(1.928961029 + 0.4387365454i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad941 \( 1 \)
good2 \( 1 + (0.964 + 0.264i)T \)
3 \( 1 + (0.359 - 0.933i)T \)
5 \( 1 + (0.480 + 0.876i)T \)
7 \( 1 + (-0.997 + 0.0667i)T \)
11 \( 1 + (0.593 + 0.805i)T \)
13 \( 1 + (-0.892 + 0.451i)T \)
17 \( 1 + (-0.979 - 0.199i)T \)
19 \( 1 + (0.100 + 0.994i)T \)
23 \( 1 + T \)
29 \( 1 + (0.231 + 0.972i)T \)
31 \( 1 + (0.695 + 0.718i)T \)
37 \( 1 + (0.480 - 0.876i)T \)
41 \( 1 + (0.964 + 0.264i)T \)
43 \( 1 + (0.991 - 0.133i)T \)
47 \( 1 + (-0.0334 - 0.999i)T \)
53 \( 1 + (0.100 - 0.994i)T \)
59 \( 1 + (-0.645 - 0.763i)T \)
61 \( 1 + (-0.997 + 0.0667i)T \)
67 \( 1 + (-0.166 + 0.986i)T \)
71 \( 1 + (0.100 + 0.994i)T \)
73 \( 1 + (-0.296 - 0.955i)T \)
79 \( 1 + (-0.824 - 0.565i)T \)
83 \( 1 + (-0.892 + 0.451i)T \)
89 \( 1 + (0.695 + 0.718i)T \)
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

−21.612097730888136310210230625286, −21.14754310442035732892245540419, −20.0763921904798693583460960874, −19.767232139165977723014678602589, −19.06061950980048583172568983375, −17.16234433258500290679633518655, −16.921464632075953006428203965965, −15.761920436722623991627413465112, −15.50578029114398903620079487165, −14.425588108394917973930498726403, −13.5060724445411855318825853564, −13.17329776408568519415614681600, −12.14621935360898320417802455947, −11.20415977482288825939068812958, −10.38188163777573632693421431091, −9.429835211383839890669471778751, −9.05533831154402458675953652229, −7.69628134280588855641703359506, −6.36400280431307470472548363496, −5.779168612885411112966544242605, −4.66463095066566503865652507528, −4.24903976769523600964098338233, −2.97426651310778143743924330684, −2.49875476333122557387717139812, −0.80973040203957841366236091486, 1.68471861991728764019895262014, 2.522846214145363643487762235821, 3.17601526033446288611548699870, 4.24570064048135903699642948775, 5.54992577082598681702439784384, 6.47762658036316236942902113700, 6.92689568949281029988725968592, 7.46075701239712110350681009377, 8.8838301454426505724238946467, 9.74746574327579000360921912736, 10.84535728123991521574762154630, 11.866906786360069210662427513189, 12.545703739843995647986874342854, 13.15034304080394139703385569160, 14.09991969996245905101828402721, 14.53620011151269555596636986408, 15.26750746143547357088653657490, 16.352653156222060031056835374627, 17.28761462007602935028383086214, 17.91333599235968602378334424907, 19.038467907870326735359823469940, 19.599512422077799206274829771402, 20.30801285575798100066165718551, 21.40020249387296170029996925300, 22.15277120492157202709951583298

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