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Properties

Label 1-236-236.155-r0-0-0
Degree $1$
Conductor $236$
Sign $-0.317 - 0.948i$
Analytic cond. $1.09597$
Root an. cond. $1.09597$
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  + (0.856 − 0.515i)3-s + (−0.370 − 0.928i)5-s + (−0.907 − 0.419i)7-s + (0.468 − 0.883i)9-s + (0.267 + 0.963i)11-s + (−0.468 − 0.883i)13-s + (−0.796 − 0.605i)15-s + (0.907 − 0.419i)17-s + (−0.976 − 0.214i)19-s + (−0.994 + 0.108i)21-s + (−0.161 − 0.986i)23-s + (−0.725 + 0.687i)25-s + (−0.0541 − 0.998i)27-s + (−0.947 − 0.319i)29-s + (0.976 − 0.214i)31-s + ⋯
L(s)  = 1  + (0.856 − 0.515i)3-s + (−0.370 − 0.928i)5-s + (−0.907 − 0.419i)7-s + (0.468 − 0.883i)9-s + (0.267 + 0.963i)11-s + (−0.468 − 0.883i)13-s + (−0.796 − 0.605i)15-s + (0.907 − 0.419i)17-s + (−0.976 − 0.214i)19-s + (−0.994 + 0.108i)21-s + (−0.161 − 0.986i)23-s + (−0.725 + 0.687i)25-s + (−0.0541 − 0.998i)27-s + (−0.947 − 0.319i)29-s + (0.976 − 0.214i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 236 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(236\)    =    \(2^{2} \cdot 59\)
Sign: $-0.317 - 0.948i$
Analytic conductor: \(1.09597\)
Root analytic conductor: \(1.09597\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{236} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 236,\ (0:\ ),\ -0.317 - 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7235209897 - 1.005236754i\)
\(L(\frac12)\) \(\approx\) \(0.7235209897 - 1.005236754i\)
\(L(1)\) \(\approx\) \(1.016786025 - 0.5332171820i\)
\(L(1)\) \(\approx\) \(1.016786025 - 0.5332171820i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
59 \( 1 \)
good3 \( 1 + (0.856 - 0.515i)T \)
5 \( 1 + (-0.370 - 0.928i)T \)
7 \( 1 + (-0.907 - 0.419i)T \)
11 \( 1 + (0.267 + 0.963i)T \)
13 \( 1 + (-0.468 - 0.883i)T \)
17 \( 1 + (0.907 - 0.419i)T \)
19 \( 1 + (-0.976 - 0.214i)T \)
23 \( 1 + (-0.161 - 0.986i)T \)
29 \( 1 + (-0.947 - 0.319i)T \)
31 \( 1 + (0.976 - 0.214i)T \)
37 \( 1 + (0.561 + 0.827i)T \)
41 \( 1 + (-0.161 + 0.986i)T \)
43 \( 1 + (0.267 - 0.963i)T \)
47 \( 1 + (-0.370 + 0.928i)T \)
53 \( 1 + (0.647 - 0.762i)T \)
61 \( 1 + (0.947 - 0.319i)T \)
67 \( 1 + (-0.561 + 0.827i)T \)
71 \( 1 + (0.370 - 0.928i)T \)
73 \( 1 + (0.994 + 0.108i)T \)
79 \( 1 + (0.856 + 0.515i)T \)
83 \( 1 + (0.0541 - 0.998i)T \)
89 \( 1 + (0.947 + 0.319i)T \)
97 \( 1 + (0.994 - 0.108i)T \)
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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

−26.35382698009733974705510929170, −25.86841794093910574557563175995, −24.93467035726409327933951499862, −23.74695045414209348971970675970, −22.67401454828272705732579683551, −21.692050846532881830837437830, −21.30582987422330910820286752751, −19.62149569482786486065291402536, −19.253001142102375625941716538727, −18.58150855328401088627586230075, −16.84907982925588376794183565985, −16.044696150801891802121718686803, −15.074186317817538897340808542660, −14.356425526304667789343321069959, −13.42859986321225513338571542658, −12.1314558789516206467562669694, −10.96943250821052391843758432695, −9.973890264333356358022136156375, −9.09813054066884100482323477346, −8.00507185354291926770322877857, −6.882099903952492390959330257633, −5.75426383501786226438434878172, −3.980766099269901947630673185638, −3.30581233535793827700684485099, −2.1966439295914578412637434022, 0.81143140336348178432606896795, 2.387000942771714436530348769948, 3.64806962543545275886289181303, 4.71349982042554086936424345884, 6.34096254644306900208194982162, 7.44788439270549922989540136034, 8.25006408944865486652744465473, 9.426770360128900385456051718687, 10.11379733608339203623684014992, 11.99411772117224943070127569409, 12.72778442648887638140231270173, 13.30663500080718152261481986925, 14.655155823441040068488465072286, 15.43323749370954239090317886757, 16.59617125615705348331536126281, 17.45594938842409398830220285780, 18.77706902803596559476776577793, 19.59320306571771947971481221549, 20.29953016251945716275101660388, 20.89721980592258313524565515374, 22.502634845787847080616401610154, 23.296030429212580255659412083807, 24.24956985356235596758019663897, 25.18899634119221227396795608526, 25.69456576090735206274014481052

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