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Properties

Label 1-367-367.101-r0-0-0
Degree $1$
Conductor $367$
Sign $0.972 - 0.234i$
Analytic cond. $1.70434$
Root an. cond. $1.70434$
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  + (0.916 − 0.400i)2-s + (0.600 + 0.799i)3-s + (0.679 − 0.733i)4-s + (0.751 − 0.660i)5-s + (0.870 + 0.492i)6-s + (0.994 − 0.102i)7-s + (0.328 − 0.944i)8-s + (−0.279 + 0.960i)9-s + (0.423 − 0.905i)10-s + (−0.640 + 0.767i)11-s + (0.994 + 0.102i)12-s + (−0.279 + 0.960i)13-s + (0.870 − 0.492i)14-s + (0.978 + 0.204i)15-s + (−0.0771 − 0.997i)16-s + (−0.376 − 0.926i)17-s + ⋯
L(s)  = 1  + (0.916 − 0.400i)2-s + (0.600 + 0.799i)3-s + (0.679 − 0.733i)4-s + (0.751 − 0.660i)5-s + (0.870 + 0.492i)6-s + (0.994 − 0.102i)7-s + (0.328 − 0.944i)8-s + (−0.279 + 0.960i)9-s + (0.423 − 0.905i)10-s + (−0.640 + 0.767i)11-s + (0.994 + 0.102i)12-s + (−0.279 + 0.960i)13-s + (0.870 − 0.492i)14-s + (0.978 + 0.204i)15-s + (−0.0771 − 0.997i)16-s + (−0.376 − 0.926i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $0.972 - 0.234i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ 0.972 - 0.234i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.069648882 - 0.3657512542i\)
\(L(\frac12)\) \(\approx\) \(3.069648882 - 0.3657512542i\)
\(L(1)\) \(\approx\) \(2.315411967 - 0.2173570314i\)
\(L(1)\) \(\approx\) \(2.315411967 - 0.2173570314i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (0.916 - 0.400i)T \)
3 \( 1 + (0.600 + 0.799i)T \)
5 \( 1 + (0.751 - 0.660i)T \)
7 \( 1 + (0.994 - 0.102i)T \)
11 \( 1 + (-0.640 + 0.767i)T \)
13 \( 1 + (-0.279 + 0.960i)T \)
17 \( 1 + (-0.376 - 0.926i)T \)
19 \( 1 + (-0.967 - 0.254i)T \)
23 \( 1 + (-0.0771 + 0.997i)T \)
29 \( 1 + (-0.988 - 0.153i)T \)
31 \( 1 + (0.679 - 0.733i)T \)
37 \( 1 + (-0.469 + 0.882i)T \)
41 \( 1 + (-0.640 - 0.767i)T \)
43 \( 1 + (-0.967 + 0.254i)T \)
47 \( 1 + (-0.998 + 0.0514i)T \)
53 \( 1 + (0.751 - 0.660i)T \)
59 \( 1 + (0.229 + 0.973i)T \)
61 \( 1 + (0.870 - 0.492i)T \)
67 \( 1 + (0.229 - 0.973i)T \)
71 \( 1 + (0.679 + 0.733i)T \)
73 \( 1 + (0.952 + 0.304i)T \)
79 \( 1 + (0.978 - 0.204i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.998 + 0.0514i)T \)
97 \( 1 + (-0.0771 + 0.997i)T \)
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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

−24.723482404088707867352058786484, −23.92841009928157479389673960018, −23.14633086241144346954056582048, −22.04053654415482045488669435279, −21.23664456879524103041959235820, −20.628346714609797341644903748223, −19.48515494632089299211842981148, −18.358065062863358219699502355158, −17.68347386408687491636478111579, −16.8228063472630119787286157597, −15.19639365321912970289877634204, −14.79641902288796421980173212303, −13.969961080145946748521204827290, −13.16464917419846034574225505753, −12.479698553677880051710950425658, −11.17997289959856094493707712730, −10.4148083860703951724644210559, −8.516354951181312368433161515832, −8.07390103283371072784150301670, −6.903078343131878129086616810710, −6.05779623372738097641699374497, −5.16823474546150810499703823869, −3.63340946579518680054296574453, −2.58908443258824853263536777842, −1.82377554858344950064818145518, 1.83649237053710859427637180480, 2.3568169853456862037597176382, 3.96873114170460928292247312889, 4.86132466712191258583061849118, 5.25442149074055210527969703411, 6.853190382608665748631050674586, 8.13349313206914175123783979047, 9.36199004765378845041192169887, 10.00515913477005766935650207859, 11.08645422106783926294302687701, 11.94902850337561507593777477283, 13.32897137492326151891590904919, 13.71444053649024993661281424474, 14.77092148078260039761102342665, 15.37576748012647087927902333532, 16.4755669419321867614328731605, 17.37478929246609891334173523265, 18.66720948119339475859078417495, 19.85432757836674812108336639101, 20.60958231092648258604674447955, 21.110749371753045020634357935713, 21.676663881057843615769590881535, 22.694149393109882019163915734465, 23.864815219721125651279816447544, 24.47595850062411285622597881730

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