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Properties

Label 1-367-367.7-r0-0-0
Degree $1$
Conductor $367$
Sign $0.794 - 0.607i$
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  + (−0.894 − 0.447i)2-s + (−0.967 + 0.254i)3-s + (0.600 + 0.799i)4-s + (0.0257 − 0.999i)5-s + (0.978 + 0.204i)6-s + (−0.784 + 0.620i)7-s + (−0.179 − 0.983i)8-s + (0.870 − 0.492i)9-s + (−0.469 + 0.882i)10-s + (−0.558 + 0.829i)11-s + (−0.784 − 0.620i)12-s + (0.870 − 0.492i)13-s + (0.978 − 0.204i)14-s + (0.229 + 0.973i)15-s + (−0.279 + 0.960i)16-s + (−0.988 + 0.153i)17-s + ⋯
L(s)  = 1  + (−0.894 − 0.447i)2-s + (−0.967 + 0.254i)3-s + (0.600 + 0.799i)4-s + (0.0257 − 0.999i)5-s + (0.978 + 0.204i)6-s + (−0.784 + 0.620i)7-s + (−0.179 − 0.983i)8-s + (0.870 − 0.492i)9-s + (−0.469 + 0.882i)10-s + (−0.558 + 0.829i)11-s + (−0.784 − 0.620i)12-s + (0.870 − 0.492i)13-s + (0.978 − 0.204i)14-s + (0.229 + 0.973i)15-s + (−0.279 + 0.960i)16-s + (−0.988 + 0.153i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $0.794 - 0.607i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ 0.794 - 0.607i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4671003980 - 0.1579901714i\)
\(L(\frac12)\) \(\approx\) \(0.4671003980 - 0.1579901714i\)
\(L(1)\) \(\approx\) \(0.4961127566 - 0.09460303308i\)
\(L(1)\) \(\approx\) \(0.4961127566 - 0.09460303308i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (-0.894 - 0.447i)T \)
3 \( 1 + (-0.967 + 0.254i)T \)
5 \( 1 + (0.0257 - 0.999i)T \)
7 \( 1 + (-0.784 + 0.620i)T \)
11 \( 1 + (-0.558 + 0.829i)T \)
13 \( 1 + (0.870 - 0.492i)T \)
17 \( 1 + (-0.988 + 0.153i)T \)
19 \( 1 + (0.994 + 0.102i)T \)
23 \( 1 + (-0.279 - 0.960i)T \)
29 \( 1 + (-0.843 + 0.536i)T \)
31 \( 1 + (0.600 + 0.799i)T \)
37 \( 1 + (0.679 + 0.733i)T \)
41 \( 1 + (-0.558 - 0.829i)T \)
43 \( 1 + (0.994 - 0.102i)T \)
47 \( 1 + (0.328 + 0.944i)T \)
53 \( 1 + (0.0257 - 0.999i)T \)
59 \( 1 + (0.751 - 0.660i)T \)
61 \( 1 + (0.978 - 0.204i)T \)
67 \( 1 + (0.751 + 0.660i)T \)
71 \( 1 + (0.600 - 0.799i)T \)
73 \( 1 + (0.423 - 0.905i)T \)
79 \( 1 + (0.229 - 0.973i)T \)
83 \( 1 + T \)
89 \( 1 + (0.328 + 0.944i)T \)
97 \( 1 + (-0.279 - 0.960i)T \)
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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

−24.7286253145360257984669973166, −23.83956440668377277936941707041, −23.16935081153768733968550158027, −22.39739275282520553586866431946, −21.37415374943199041211568042702, −20.0488974195405708054948522522, −19.0506396264499605614687168370, −18.502273725295261819579368459536, −17.77681171899625212215118489596, −16.82720624317471689332166364093, −15.97005160320466503104548418128, −15.47698047093661518494453748556, −13.867041845462552509340006743433, −13.317856985013710238856957787550, −11.458480956435790237886884281180, −11.17344088871693116706947690568, −10.207088756891449027631464860304, −9.39209445095151407410119629106, −7.84884642882903208230555421174, −7.0756988252223063737387605693, −6.27523123278656656227342206794, −5.61478445289792779842609868294, −3.84977113442073268432524899294, −2.37277455880934682529123030535, −0.8041115768283656954586691463, 0.69571804819824917033806680686, 2.05844733237431730532836076721, 3.5757664549672498246615333472, 4.816903814996583906589030085879, 5.92772153184000786368604019892, 6.92404879797916425389397490869, 8.22619993209994988167360098267, 9.18978845553794816156758245246, 9.9298928695141777453580176143, 10.84380605842695562962549156976, 11.90359500629770511833470841353, 12.629746346324555809868073979938, 13.16426845862831582842017483157, 15.4700789884182421331993421943, 15.90211506619275430875328131221, 16.59690582955471812786764536895, 17.70767697496387217911419600328, 18.13345375997510635254532060468, 19.18890994733756110545888788687, 20.48729146337739851628253542675, 20.67537699449777171887781139345, 22.01121486115677959009217739470, 22.54324478783748772783575436503, 23.77563033743408131148680844882, 24.69263383261943593991666245038

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