↑

Properties

Label 1-317-317.123-r0-0-0
Degree $1$
Conductor $317$
Sign $0.271 + 0.962i$
Analytic cond. $1.47214$
Root an. cond. $1.47214$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.368 − 0.929i)2-s + (0.177 − 0.984i)3-s + (−0.727 + 0.685i)4-s + (−0.641 − 0.767i)5-s + (−0.980 + 0.197i)6-s + (−0.999 + 0.0397i)7-s + (0.905 + 0.423i)8-s + (−0.936 − 0.350i)9-s + (−0.476 + 0.878i)10-s + (−0.405 − 0.914i)11-s + (0.545 + 0.838i)12-s + (0.255 − 0.966i)13-s + (0.405 + 0.914i)14-s + (−0.869 + 0.494i)15-s + (0.0596 − 0.998i)16-s + (−0.921 + 0.387i)17-s + ⋯
L(s)  = 1  + (−0.368 − 0.929i)2-s + (0.177 − 0.984i)3-s + (−0.727 + 0.685i)4-s + (−0.641 − 0.767i)5-s + (−0.980 + 0.197i)6-s + (−0.999 + 0.0397i)7-s + (0.905 + 0.423i)8-s + (−0.936 − 0.350i)9-s + (−0.476 + 0.878i)10-s + (−0.405 − 0.914i)11-s + (0.545 + 0.838i)12-s + (0.255 − 0.966i)13-s + (0.405 + 0.914i)14-s + (−0.869 + 0.494i)15-s + (0.0596 − 0.998i)16-s + (−0.921 + 0.387i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(317\)
Sign: $0.271 + 0.962i$
Analytic conductor: \(1.47214\)
Root analytic conductor: \(1.47214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{317} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 317,\ (0:\ ),\ 0.271 + 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1965898824 - 0.1487753979i\)
\(L(\frac12)\) \(\approx\) \(-0.1965898824 - 0.1487753979i\)
\(L(1)\) \(\approx\) \(0.3046723019 - 0.4506795022i\)
\(L(1)\) \(\approx\) \(0.3046723019 - 0.4506795022i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad317 \( 1 \)
good2 \( 1 + (-0.368 - 0.929i)T \)
3 \( 1 + (0.177 - 0.984i)T \)
5 \( 1 + (-0.641 - 0.767i)T \)
7 \( 1 + (-0.999 + 0.0397i)T \)
11 \( 1 + (-0.405 - 0.914i)T \)
13 \( 1 + (0.255 - 0.966i)T \)
17 \( 1 + (-0.921 + 0.387i)T \)
19 \( 1 + (0.980 + 0.197i)T \)
23 \( 1 + (-0.961 + 0.274i)T \)
29 \( 1 + (0.999 + 0.0397i)T \)
31 \( 1 + (0.511 + 0.859i)T \)
37 \( 1 + (0.293 + 0.955i)T \)
41 \( 1 + (-0.754 - 0.656i)T \)
43 \( 1 + (-0.0992 + 0.995i)T \)
47 \( 1 + (-0.511 - 0.859i)T \)
53 \( 1 + (-0.980 - 0.197i)T \)
59 \( 1 + (-0.0992 - 0.995i)T \)
61 \( 1 + (-0.992 - 0.119i)T \)
67 \( 1 + (-0.780 - 0.625i)T \)
71 \( 1 + (0.869 + 0.494i)T \)
73 \( 1 + (0.804 - 0.594i)T \)
79 \( 1 + (-0.727 - 0.685i)T \)
83 \( 1 + (0.971 + 0.236i)T \)
89 \( 1 + (0.368 + 0.929i)T \)
97 \( 1 + (-0.293 - 0.955i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.09472893139720933144250689919, −25.380632340710253354143859306345, −24.07163127382533878762237582722, −23.02786797233447078630063823409, −22.56929126950716902049841508092, −21.78800167180188297193365354086, −20.22794170813362511631739761307, −19.609915838512588404903457535511, −18.59589745201360505432253777410, −17.722663259100181792046149850494, −16.47439271964081762095147674652, −15.80220705441856117748279453398, −15.3720130703324748791678574321, −14.28381363533971641567923152754, −13.53377682853847855374624942169, −11.91821069545702850805939869788, −10.728027915516521284819890313604, −9.86030061313490828134985456304, −9.21053807894409622898764205531, −7.9945946985043181193467433686, −6.97169659234756332979377722360, −6.13614674042349643088830316572, −4.68137735691042067080410846733, −3.93557607334557864305046078223, −2.579989875171668713687279322418, 0.18562081635461822668270012493, 1.33136668268462649405863069694, 2.90441654505171358758118551219, 3.55870876115182869002419622140, 5.16825170084259670514243548986, 6.46749922097750174044180326083, 7.94491744148919600847722823342, 8.350896805033666974752032483771, 9.40974211215050112365517323090, 10.62971240697318464813942659117, 11.75592913887100088690125210603, 12.41923538644149538473523432011, 13.2605280054925881420240221664, 13.76690750785554787460835095662, 15.60820197538587769552365867124, 16.43463279195780668205192914805, 17.55432509524996245464923190995, 18.37179274137283004050268838249, 19.28001670720217057504712942725, 19.887325980006843385145512762859, 20.42417435001697505104037310511, 21.746663738208225705481644675, 22.72623287936427957007362778819, 23.4847189238639114240706414544, 24.46762548776354947323620056873

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