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Properties

Label 1-367-367.162-r0-0-0
Degree $1$
Conductor $367$
Sign $0.999 + 0.00982i$
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  + (−0.344 + 0.938i)2-s + (−0.376 + 0.926i)3-s + (−0.762 − 0.647i)4-s + (0.679 − 0.733i)5-s + (−0.739 − 0.672i)6-s + (−0.843 + 0.536i)7-s + (0.870 − 0.492i)8-s + (−0.716 − 0.697i)9-s + (0.454 + 0.890i)10-s + (−0.586 − 0.810i)11-s + (0.886 − 0.462i)12-s + (−0.246 + 0.969i)13-s + (−0.212 − 0.977i)14-s + (0.423 + 0.905i)15-s + (0.162 + 0.986i)16-s + (−0.957 + 0.287i)17-s + ⋯
L(s)  = 1  + (−0.344 + 0.938i)2-s + (−0.376 + 0.926i)3-s + (−0.762 − 0.647i)4-s + (0.679 − 0.733i)5-s + (−0.739 − 0.672i)6-s + (−0.843 + 0.536i)7-s + (0.870 − 0.492i)8-s + (−0.716 − 0.697i)9-s + (0.454 + 0.890i)10-s + (−0.586 − 0.810i)11-s + (0.886 − 0.462i)12-s + (−0.246 + 0.969i)13-s + (−0.212 − 0.977i)14-s + (0.423 + 0.905i)15-s + (0.162 + 0.986i)16-s + (−0.957 + 0.287i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $0.999 + 0.00982i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (162, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ 0.999 + 0.00982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6017691865 + 0.002957022845i\)
\(L(\frac12)\) \(\approx\) \(0.6017691865 + 0.002957022845i\)
\(L(1)\) \(\approx\) \(0.6107977794 + 0.2567948667i\)
\(L(1)\) \(\approx\) \(0.6107977794 + 0.2567948667i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (-0.344 + 0.938i)T \)
3 \( 1 + (-0.376 + 0.926i)T \)
5 \( 1 + (0.679 - 0.733i)T \)
7 \( 1 + (-0.843 + 0.536i)T \)
11 \( 1 + (-0.586 - 0.810i)T \)
13 \( 1 + (-0.246 + 0.969i)T \)
17 \( 1 + (-0.957 + 0.287i)T \)
19 \( 1 + (0.360 - 0.932i)T \)
23 \( 1 + (0.162 - 0.986i)T \)
29 \( 1 + (0.751 - 0.660i)T \)
31 \( 1 + (0.941 - 0.336i)T \)
37 \( 1 + (0.653 + 0.756i)T \)
41 \( 1 + (-0.408 - 0.912i)T \)
43 \( 1 + (0.627 - 0.778i)T \)
47 \( 1 + (-0.279 - 0.960i)T \)
53 \( 1 + (0.295 + 0.955i)T \)
59 \( 1 + (-0.469 - 0.882i)T \)
61 \( 1 + (-0.739 + 0.672i)T \)
67 \( 1 + (-0.529 - 0.848i)T \)
71 \( 1 + (0.941 + 0.336i)T \)
73 \( 1 + (-0.923 + 0.384i)T \)
79 \( 1 + (0.572 + 0.819i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (0.971 + 0.238i)T \)
97 \( 1 + (0.773 + 0.633i)T \)
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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

−24.96549671141276486615841084097, −23.333348457042105540291542060090, −22.8006621947000166561637738124, −22.24448569202731315362153143220, −21.09295257821924867005556060968, −19.970769381286095318976514417932, −19.48195355592562496946386450323, −18.34983467713409517167840488287, −17.84630569733645862664890287945, −17.237734453757850801396532156354, −15.9961153955182490709559594153, −14.455403123306306258904036586061, −13.41768441548754708734300418551, −13.00410091564896914442764403379, −12.073174906846037012847941194323, −10.90017905320716382538501149371, −10.24682148546377819605861586629, −9.433594579076831552228191200381, −7.90026742347954431671073961362, −7.19410351432628287237514964201, −6.09708076933113027051767057078, −4.87141687574468264520271197649, −3.1925356592653953662122280094, −2.47741361798103459064542675213, −1.24874518164626755188633265780, 0.46897962480682011847366707166, 2.56731087320651817640181025666, 4.27489338635553558942434291596, 5.04935232354372192296420698325, 6.039059358335542362033341568874, 6.62924763144772219617402444003, 8.49650674684562808092622008099, 9.01132420184849259093140509978, 9.786259463935485013343821603775, 10.67686485463492880880743496661, 12.02232962090676139470144672197, 13.293772556915684511910381490421, 13.94289638954206259843079603580, 15.32281092962136300890992777498, 15.83444561050628283464253625891, 16.656827618833735597849064945932, 17.215087869229051307007548226756, 18.26142689339584957600082716057, 19.23770592849462052097173469533, 20.29988350623679482830864756393, 21.57042006317709561205139106460, 21.94081358259269261996968258659, 22.93939698977288095643733548676, 24.04333055317816928803459035385, 24.5863544527448710368420223405

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