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Properties

Label 1-367-367.46-r0-0-0
Degree $1$
Conductor $367$
Sign $-0.0822 + 0.996i$
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  + (0.751 + 0.660i)2-s + (−0.998 + 0.0514i)3-s + (0.128 + 0.991i)4-s + (0.952 − 0.304i)5-s + (−0.784 − 0.620i)6-s + (−0.179 + 0.983i)7-s + (−0.558 + 0.829i)8-s + (0.994 − 0.102i)9-s + (0.916 + 0.400i)10-s + (0.679 − 0.733i)11-s + (−0.179 − 0.983i)12-s + (0.994 − 0.102i)13-s + (−0.784 + 0.620i)14-s + (−0.935 + 0.352i)15-s + (−0.967 + 0.254i)16-s + (−0.279 + 0.960i)17-s + ⋯
L(s)  = 1  + (0.751 + 0.660i)2-s + (−0.998 + 0.0514i)3-s + (0.128 + 0.991i)4-s + (0.952 − 0.304i)5-s + (−0.784 − 0.620i)6-s + (−0.179 + 0.983i)7-s + (−0.558 + 0.829i)8-s + (0.994 − 0.102i)9-s + (0.916 + 0.400i)10-s + (0.679 − 0.733i)11-s + (−0.179 − 0.983i)12-s + (0.994 − 0.102i)13-s + (−0.784 + 0.620i)14-s + (−0.935 + 0.352i)15-s + (−0.967 + 0.254i)16-s + (−0.279 + 0.960i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0822 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0822 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $-0.0822 + 0.996i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ -0.0822 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.162562084 + 1.262464083i\)
\(L(\frac12)\) \(\approx\) \(1.162562084 + 1.262464083i\)
\(L(1)\) \(\approx\) \(1.205570826 + 0.7044377123i\)
\(L(1)\) \(\approx\) \(1.205570826 + 0.7044377123i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (0.751 + 0.660i)T \)
3 \( 1 + (-0.998 + 0.0514i)T \)
5 \( 1 + (0.952 - 0.304i)T \)
7 \( 1 + (-0.179 + 0.983i)T \)
11 \( 1 + (0.679 - 0.733i)T \)
13 \( 1 + (0.994 - 0.102i)T \)
17 \( 1 + (-0.279 + 0.960i)T \)
19 \( 1 + (0.328 - 0.944i)T \)
23 \( 1 + (-0.967 - 0.254i)T \)
29 \( 1 + (0.870 + 0.492i)T \)
31 \( 1 + (0.128 + 0.991i)T \)
37 \( 1 + (-0.894 + 0.447i)T \)
41 \( 1 + (0.679 + 0.733i)T \)
43 \( 1 + (0.328 + 0.944i)T \)
47 \( 1 + (-0.640 - 0.767i)T \)
53 \( 1 + (0.952 - 0.304i)T \)
59 \( 1 + (-0.716 + 0.697i)T \)
61 \( 1 + (-0.784 + 0.620i)T \)
67 \( 1 + (-0.716 - 0.697i)T \)
71 \( 1 + (0.128 - 0.991i)T \)
73 \( 1 + (0.514 + 0.857i)T \)
79 \( 1 + (-0.935 - 0.352i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.640 - 0.767i)T \)
97 \( 1 + (-0.967 - 0.254i)T \)
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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

−24.23819928984867368956107736592, −23.07646807029429653893064285202, −22.77776481499426102371590042694, −22.02138801951507155073173116304, −20.924950823721338458709208231242, −20.444669057882230737866968844953, −19.16012547452714807987735697365, −18.18034884429126553342531259942, −17.52522088982072348764457653388, −16.447692180946283337617862180370, −15.53656266022279897005318173090, −14.063159207661754711245253644286, −13.745510130929026542480628030190, −12.657977470624347613605949833154, −11.77323216833338722227439666105, −10.84827218848993460485961800397, −10.10156476358447188629514495792, −9.4468565545219319152634353047, −7.27295385254555217971535656633, −6.40339108501831798074652805043, −5.7234521511987063101788821700, −4.51588778087078763111969567662, −3.68755784434255132467921033606, −2.02933799699352393303377952805, −1.05818786020649509759403576573, 1.55612706074110754183968794327, 3.09332465995227904598672737966, 4.4446513990572377286537995966, 5.44570669709065247875827782406, 6.17794568938023485431082038575, 6.60152386619047566957708137298, 8.39864233922136275777192488903, 9.10584789400315477252931733151, 10.52295821436816054322713577244, 11.58234309783399111456184027630, 12.39576213730851011909357469334, 13.210144054407762900173605340850, 14.039114969408034418022427849456, 15.256738133153970140573479419475, 16.10644094130203962106842458988, 16.69337844003016219192453622431, 17.848888357700684040822090421665, 18.09979738587552856651464496334, 19.67777003066802234780612522404, 21.22820144777410985240429770084, 21.578131030058791074331089170718, 22.21556400828984567669144852850, 23.08232689294810959235923433013, 24.27176735253928162139021733809, 24.50679472821104331632816621995

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