Properties

Degree 1
Conductor 367
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s − 11-s − 12-s + 13-s + 14-s + 15-s + 16-s − 17-s + 18-s − 19-s − 20-s − 21-s − 22-s + 23-s − 24-s + 25-s + 26-s − 27-s + 28-s + ⋯
L(s,χ)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s − 11-s − 12-s + 13-s + 14-s + 15-s + 16-s − 17-s + 18-s − 19-s − 20-s − 21-s − 22-s + 23-s − 24-s + 25-s + 26-s − 27-s + 28-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(367\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{367} (366, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(1,\ 367,\ (1:\ ),\ 1)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.614155034$
$L(\frac12,\chi)$  $\approx$  $2.614155034$
$L(\chi,1)$  $\approx$  1.475908214
$L(1,\chi)$  $\approx$  1.475908214

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−24.07295344427227916874467011058, −23.39375800551083146116290828474, −23.04614205162520113168926827583, −21.91466984398112239197876735766, −21.05500027791353624261487685782, −20.46830595644090939713749872424, −19.1496905173913490427105806983, −18.27755729397218562027159172359, −17.18096190396635690138152398530, −16.219261881409954433527728838799, −15.43372193075151330707375233747, −14.89494636863222701186534308176, −13.36101839846742971101565702138, −12.78819201979949167757426663937, −11.61959262889186169649375956399, −11.11011596746567091634282514029, −10.554050395696314949375444037175, −8.4843781011859145068327343938, −7.53450453793685223985425491642, −6.60156249957092905297031518942, −5.48162804846609629609268813022, −4.610836189071464547860425879469, −3.929459366410916690305532924, −2.32486240416487945515243663933, −0.862638631695890076901632988102, 0.862638631695890076901632988102, 2.32486240416487945515243663933, 3.929459366410916690305532924, 4.610836189071464547860425879469, 5.48162804846609629609268813022, 6.60156249957092905297031518942, 7.53450453793685223985425491642, 8.4843781011859145068327343938, 10.554050395696314949375444037175, 11.11011596746567091634282514029, 11.61959262889186169649375956399, 12.78819201979949167757426663937, 13.36101839846742971101565702138, 14.89494636863222701186534308176, 15.43372193075151330707375233747, 16.219261881409954433527728838799, 17.18096190396635690138152398530, 18.27755729397218562027159172359, 19.1496905173913490427105806983, 20.46830595644090939713749872424, 21.05500027791353624261487685782, 21.91466984398112239197876735766, 23.04614205162520113168926827583, 23.39375800551083146116290828474, 24.07295344427227916874467011058

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