Properties

Label 1-367-367.59-r0-0-0
Degree $1$
Conductor $367$
Sign $0.986 + 0.166i$
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  + (−0.967 − 0.254i)2-s + (0.978 + 0.204i)3-s + (0.870 + 0.492i)4-s + (0.328 + 0.944i)5-s + (−0.894 − 0.447i)6-s + (0.751 − 0.660i)7-s + (−0.716 − 0.697i)8-s + (0.916 + 0.400i)9-s + (−0.0771 − 0.997i)10-s + (−0.988 − 0.153i)11-s + (0.751 + 0.660i)12-s + (0.916 + 0.400i)13-s + (−0.894 + 0.447i)14-s + (0.128 + 0.991i)15-s + (0.514 + 0.857i)16-s + (0.423 − 0.905i)17-s + ⋯
L(s)  = 1  + (−0.967 − 0.254i)2-s + (0.978 + 0.204i)3-s + (0.870 + 0.492i)4-s + (0.328 + 0.944i)5-s + (−0.894 − 0.447i)6-s + (0.751 − 0.660i)7-s + (−0.716 − 0.697i)8-s + (0.916 + 0.400i)9-s + (−0.0771 − 0.997i)10-s + (−0.988 − 0.153i)11-s + (0.751 + 0.660i)12-s + (0.916 + 0.400i)13-s + (−0.894 + 0.447i)14-s + (0.128 + 0.991i)15-s + (0.514 + 0.857i)16-s + (0.423 − 0.905i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $0.986 + 0.166i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ 0.986 + 0.166i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.394752861 + 0.1168283454i\)
\(L(\frac12)\) \(\approx\) \(1.394752861 + 0.1168283454i\)
\(L(1)\) \(\approx\) \(1.124657332 + 0.04863271685i\)
\(L(1)\) \(\approx\) \(1.124657332 + 0.04863271685i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (-0.967 - 0.254i)T \)
3 \( 1 + (0.978 + 0.204i)T \)
5 \( 1 + (0.328 + 0.944i)T \)
7 \( 1 + (0.751 - 0.660i)T \)
11 \( 1 + (-0.988 - 0.153i)T \)
13 \( 1 + (0.916 + 0.400i)T \)
17 \( 1 + (0.423 - 0.905i)T \)
19 \( 1 + (0.229 - 0.973i)T \)
23 \( 1 + (0.514 - 0.857i)T \)
29 \( 1 + (-0.469 - 0.882i)T \)
31 \( 1 + (0.870 + 0.492i)T \)
37 \( 1 + (-0.279 + 0.960i)T \)
41 \( 1 + (-0.988 + 0.153i)T \)
43 \( 1 + (0.229 + 0.973i)T \)
47 \( 1 + (-0.935 + 0.352i)T \)
53 \( 1 + (0.328 + 0.944i)T \)
59 \( 1 + (-0.998 + 0.0514i)T \)
61 \( 1 + (-0.894 + 0.447i)T \)
67 \( 1 + (-0.998 - 0.0514i)T \)
71 \( 1 + (0.870 - 0.492i)T \)
73 \( 1 + (-0.558 + 0.829i)T \)
79 \( 1 + (0.128 - 0.991i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.935 + 0.352i)T \)
97 \( 1 + (0.514 - 0.857i)T \)
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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

−24.89058283733854816623013374791, −24.10724730026026581070223523224, −23.42845614501986196057276642294, −21.33911467344715647842823087771, −20.94593390766239925952345144208, −20.31357821781547533515905804240, −19.24839122580298092187953823291, −18.40219803196970349755923608586, −17.812839560892370030562681170907, −16.69966480031504465784226886297, −15.6498860954091543243453917494, −15.12208692720178491210911706454, −13.99842042136796458047370713285, −12.927845176970512151275606984257, −12.058742978312956028772530612912, −10.69240339627072685769743821589, −9.770194242687117171979886545996, −8.76021771165671729781710731522, −8.25035750436457629610253564521, −7.55906317976453410062661457540, −5.97138632921886052735241351909, −5.15112015383678533195810614157, −3.42620775446542287140523957963, −2.008642537343646887118527743871, −1.36699804804665318183323040721, 1.34669066394399742275181247937, 2.590106854361508927214366617399, 3.238168556750942749393361753964, 4.68585050168500032495103358080, 6.49157258249462172828257146017, 7.42273694493813826314369361943, 8.10224356613334494732975029661, 9.125137743096227399432364431833, 10.13908856270340165551681746928, 10.76064261077493477501229321660, 11.59827164164982410720981453412, 13.299112228966957656996234943883, 13.87806784856923358964543004103, 15.02120317868863171422523311302, 15.73926613923341534177305632859, 16.78881765898724841524380490783, 18.01181839513634423407459524043, 18.4856642100590967671273545827, 19.27025582346061429557134871881, 20.40376348119383985377618578508, 20.95834846770721983211621154280, 21.5489742399272817784293422909, 22.935941504766344554103244455538, 24.12353457692294965872821970067, 24.975547345146305961722726527973

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