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Properties

Label 1-417-417.371-r0-0-0
Degree $1$
Conductor $417$
Sign $0.356 + 0.934i$
Analytic cond. $1.93653$
Root an. cond. $1.93653$
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  + (0.291 + 0.956i)2-s + (−0.829 + 0.557i)4-s + (0.949 − 0.313i)5-s + (0.613 + 0.789i)7-s + (−0.775 − 0.631i)8-s + (0.576 + 0.816i)10-s + (0.877 − 0.480i)11-s + (0.995 − 0.0909i)13-s + (−0.576 + 0.816i)14-s + (0.377 − 0.926i)16-s + (−0.247 − 0.968i)17-s + (0.715 − 0.699i)19-s + (−0.613 + 0.789i)20-s + (0.715 + 0.699i)22-s + (−0.990 + 0.136i)23-s + ⋯
L(s)  = 1  + (0.291 + 0.956i)2-s + (−0.829 + 0.557i)4-s + (0.949 − 0.313i)5-s + (0.613 + 0.789i)7-s + (−0.775 − 0.631i)8-s + (0.576 + 0.816i)10-s + (0.877 − 0.480i)11-s + (0.995 − 0.0909i)13-s + (−0.576 + 0.816i)14-s + (0.377 − 0.926i)16-s + (−0.247 − 0.968i)17-s + (0.715 − 0.699i)19-s + (−0.613 + 0.789i)20-s + (0.715 + 0.699i)22-s + (−0.990 + 0.136i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 417 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 417 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(417\)    =    \(3 \cdot 139\)
Sign: $0.356 + 0.934i$
Analytic conductor: \(1.93653\)
Root analytic conductor: \(1.93653\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{417} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 417,\ (0:\ ),\ 0.356 + 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.536400103 + 1.058011493i\)
\(L(\frac12)\) \(\approx\) \(1.536400103 + 1.058011493i\)
\(L(1)\) \(\approx\) \(1.295369344 + 0.6551878028i\)
\(L(1)\) \(\approx\) \(1.295369344 + 0.6551878028i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
139 \( 1 \)
good2 \( 1 + (0.291 + 0.956i)T \)
5 \( 1 + (0.949 - 0.313i)T \)
7 \( 1 + (0.613 + 0.789i)T \)
11 \( 1 + (0.877 - 0.480i)T \)
13 \( 1 + (0.995 - 0.0909i)T \)
17 \( 1 + (-0.247 - 0.968i)T \)
19 \( 1 + (0.715 - 0.699i)T \)
23 \( 1 + (-0.990 + 0.136i)T \)
29 \( 1 + (-0.898 - 0.439i)T \)
31 \( 1 + (-0.648 + 0.761i)T \)
37 \( 1 + (0.113 + 0.993i)T \)
41 \( 1 + (0.998 - 0.0455i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.746 + 0.665i)T \)
53 \( 1 + (-0.974 - 0.225i)T \)
59 \( 1 + (-0.0682 + 0.997i)T \)
61 \( 1 + (0.998 + 0.0455i)T \)
67 \( 1 + (0.0227 + 0.999i)T \)
71 \( 1 + (-0.746 - 0.665i)T \)
73 \( 1 + (-0.934 + 0.356i)T \)
79 \( 1 + (0.460 + 0.887i)T \)
83 \( 1 + (-0.0227 + 0.999i)T \)
89 \( 1 + (-0.803 - 0.595i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−23.92755122336234024313234526122, −22.97148695946651059913660887727, −22.24766540446111977854499467613, −21.439386172276589168351796889806, −20.58963560218209830069185287878, −20.07139992467182620419637212717, −18.912916532484214092691788682566, −17.961900492267184880653791141194, −17.5025398639162813645858618366, −16.38451345698602403032839873455, −14.67375736737945608220738646395, −14.35291837262436357768106671400, −13.42139124656347243184170515382, −12.63942139277158421072795818916, −11.39169668119443788501687070199, −10.769844590117983515025316271082, −9.86291331993989931734470398980, −9.0764176928881131168792241748, −7.82935291748008645462935964975, −6.38276939630022516831934729537, −5.587065116729974585366129948128, −4.23787551852147424910413315680, −3.56773483023752377884821695000, −1.94083583331134579755458248127, −1.39192012000526699711014995668, 1.31384473593128495148451704108, 2.83159475374492964772624188849, 4.187494986267843883716402625335, 5.33302673079636717487444613605, 5.89599545425070105556769302326, 6.8747183367021473924957731624, 8.16354333174344621177880330447, 9.01039872565030519014560764911, 9.54419582834429445429945119660, 11.2035947348094983818747293599, 12.10587314938314250701207070392, 13.26284289949676327518097669986, 13.92006183425685709545094411147, 14.62143417168758888971547586318, 15.79669168328084744777660725542, 16.36949657212617430958555934325, 17.55669742936131128217735485293, 17.974961628590258869726472949503, 18.833512441938094576742928183882, 20.37776623744708288570925793119, 21.16991613128296778617233533539, 22.08055220578938966813369411153, 22.4677273717688265782110563671, 23.94459975072552343320253425225, 24.40480965180713235085479664433

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