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Properties

Label 1-367-367.9-r0-0-0
Degree $1$
Conductor $367$
Sign $-0.117 + 0.993i$
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  + (−0.469 − 0.882i)2-s + (−0.0771 − 0.997i)3-s + (−0.558 + 0.829i)4-s + (−0.894 + 0.447i)5-s + (−0.843 + 0.536i)6-s + (0.870 − 0.492i)7-s + (0.994 + 0.102i)8-s + (−0.988 + 0.153i)9-s + (0.815 + 0.579i)10-s + (0.328 − 0.944i)11-s + (0.870 + 0.492i)12-s + (−0.988 + 0.153i)13-s + (−0.843 − 0.536i)14-s + (0.514 + 0.857i)15-s + (−0.376 − 0.926i)16-s + (−0.935 + 0.352i)17-s + ⋯
L(s)  = 1  + (−0.469 − 0.882i)2-s + (−0.0771 − 0.997i)3-s + (−0.558 + 0.829i)4-s + (−0.894 + 0.447i)5-s + (−0.843 + 0.536i)6-s + (0.870 − 0.492i)7-s + (0.994 + 0.102i)8-s + (−0.988 + 0.153i)9-s + (0.815 + 0.579i)10-s + (0.328 − 0.944i)11-s + (0.870 + 0.492i)12-s + (−0.988 + 0.153i)13-s + (−0.843 − 0.536i)14-s + (0.514 + 0.857i)15-s + (−0.376 − 0.926i)16-s + (−0.935 + 0.352i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $-0.117 + 0.993i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ -0.117 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09682357120 - 0.1089097047i\)
\(L(\frac12)\) \(\approx\) \(-0.09682357120 - 0.1089097047i\)
\(L(1)\) \(\approx\) \(0.3935020508 - 0.3498590736i\)
\(L(1)\) \(\approx\) \(0.3935020508 - 0.3498590736i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (-0.469 - 0.882i)T \)
3 \( 1 + (-0.0771 - 0.997i)T \)
5 \( 1 + (-0.894 + 0.447i)T \)
7 \( 1 + (0.870 - 0.492i)T \)
11 \( 1 + (0.328 - 0.944i)T \)
13 \( 1 + (-0.988 + 0.153i)T \)
17 \( 1 + (-0.935 + 0.352i)T \)
19 \( 1 + (-0.279 - 0.960i)T \)
23 \( 1 + (-0.376 + 0.926i)T \)
29 \( 1 + (-0.716 - 0.697i)T \)
31 \( 1 + (-0.558 + 0.829i)T \)
37 \( 1 + (-0.640 - 0.767i)T \)
41 \( 1 + (0.328 + 0.944i)T \)
43 \( 1 + (-0.279 + 0.960i)T \)
47 \( 1 + (-0.967 + 0.254i)T \)
53 \( 1 + (-0.894 + 0.447i)T \)
59 \( 1 + (0.916 + 0.400i)T \)
61 \( 1 + (-0.843 - 0.536i)T \)
67 \( 1 + (0.916 - 0.400i)T \)
71 \( 1 + (-0.558 - 0.829i)T \)
73 \( 1 + (0.0257 + 0.999i)T \)
79 \( 1 + (0.514 - 0.857i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.967 + 0.254i)T \)
97 \( 1 + (-0.376 + 0.926i)T \)
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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

−25.28797801559056956823517406230, −24.45401479803039025535079959297, −23.76938992717833056527053521523, −22.56650730977669895174774443044, −22.23732713614758307382286190141, −20.61829548722439356557000875739, −20.19806264875475559759032420, −19.090782162770249736134068185650, −17.98459875924929564830455807360, −17.12245441188514173424815239075, −16.44549286431617394380925115364, −15.42594639446789690305590663724, −14.93856345135358971255934136748, −14.29177108377959815655766631534, −12.57211571669238764509114038627, −11.61220109765095256783738964085, −10.59464408366082671839085317366, −9.568858382120226505560127355929, −8.72836543007777133984243696919, −7.97802089878902419198728779959, −6.91652232173677030101900431516, −5.42877905582866793605295958495, −4.74170671325838008548901972786, −4.00653801259713574420142679781, −2.00520624787433889387860272275, 0.104872377789141416294891690883, 1.51541920636064595500307194928, 2.638752762146283356740800111970, 3.7797838689553866902141946098, 4.92045736203692417108082330648, 6.67033946119615776843182719374, 7.5927001906881053153864191439, 8.19852541956956666154969166904, 9.22490170922171674246163846649, 10.88537130460188447066143654105, 11.245711615582274229819161457688, 11.991935730498207384172784558474, 13.06217288631800953648382347595, 13.9350361185004823949072081129, 14.82120643046549245090856488838, 16.3401242425556255843632068427, 17.40289554283665744402788013973, 17.86929529436504130904305183759, 18.95107761261822176069173346929, 19.66229905403447421149290421069, 19.97150663975380940757135395086, 21.428145596092434064290790566666, 22.1740657002238052922568806883, 23.19290606775195606456803950328, 24.01639434737662602383030543601

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