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Properties

Label 1-235-235.4-r0-0-0
Degree $1$
Conductor $235$
Sign $-0.213 + 0.976i$
Analytic cond. $1.09133$
Root an. cond. $1.09133$
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  + (−0.854 − 0.519i)2-s + (0.576 + 0.816i)3-s + (0.460 + 0.887i)4-s + (−0.0682 − 0.997i)6-s + (−0.962 − 0.269i)7-s + (0.0682 − 0.997i)8-s + (−0.334 + 0.942i)9-s + (−0.990 + 0.136i)11-s + (−0.460 + 0.887i)12-s + (0.775 + 0.631i)13-s + (0.682 + 0.730i)14-s + (−0.576 + 0.816i)16-s + (0.990 + 0.136i)17-s + (0.775 − 0.631i)18-s + (0.203 + 0.979i)19-s + ⋯
L(s)  = 1  + (−0.854 − 0.519i)2-s + (0.576 + 0.816i)3-s + (0.460 + 0.887i)4-s + (−0.0682 − 0.997i)6-s + (−0.962 − 0.269i)7-s + (0.0682 − 0.997i)8-s + (−0.334 + 0.942i)9-s + (−0.990 + 0.136i)11-s + (−0.460 + 0.887i)12-s + (0.775 + 0.631i)13-s + (0.682 + 0.730i)14-s + (−0.576 + 0.816i)16-s + (0.990 + 0.136i)17-s + (0.775 − 0.631i)18-s + (0.203 + 0.979i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $-0.213 + 0.976i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ -0.213 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4105575027 + 0.5101776859i\)
\(L(\frac12)\) \(\approx\) \(0.4105575027 + 0.5101776859i\)
\(L(1)\) \(\approx\) \(0.6760346533 + 0.2003775059i\)
\(L(1)\) \(\approx\) \(0.6760346533 + 0.2003775059i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
47 \( 1 \)
good2 \( 1 + (-0.854 - 0.519i)T \)
3 \( 1 + (0.576 + 0.816i)T \)
7 \( 1 + (-0.962 - 0.269i)T \)
11 \( 1 + (-0.990 + 0.136i)T \)
13 \( 1 + (0.775 + 0.631i)T \)
17 \( 1 + (0.990 + 0.136i)T \)
19 \( 1 + (0.203 + 0.979i)T \)
23 \( 1 + (-0.854 + 0.519i)T \)
29 \( 1 + (-0.775 + 0.631i)T \)
31 \( 1 + (-0.576 + 0.816i)T \)
37 \( 1 + (-0.682 + 0.730i)T \)
41 \( 1 + (-0.0682 - 0.997i)T \)
43 \( 1 + (-0.460 - 0.887i)T \)
53 \( 1 + (0.0682 + 0.997i)T \)
59 \( 1 + (0.460 - 0.887i)T \)
61 \( 1 + (0.682 + 0.730i)T \)
67 \( 1 + (-0.962 + 0.269i)T \)
71 \( 1 + (0.854 - 0.519i)T \)
73 \( 1 + (0.334 + 0.942i)T \)
79 \( 1 + (-0.917 + 0.398i)T \)
83 \( 1 + (0.990 - 0.136i)T \)
89 \( 1 + (0.203 - 0.979i)T \)
97 \( 1 + (0.576 + 0.816i)T \)
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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

−26.0092167717261594322049968716, −25.29456979825325648696758567881, −24.37074306320113930006017916016, −23.52177541495612976009717247046, −22.691262906196841171502269912153, −20.97269901153075773164746112935, −20.12187979527944540921070874798, −19.24750523614979477166979892059, −18.454601628429595852044572006900, −17.91190829027938851652402068742, −16.53348655354525176344936322234, −15.69719741215455257613550873377, −14.81473159934232273415373208084, −13.5725338904529309650988367780, −12.81870930399121214806566592012, −11.48621749009521147263194897570, −10.18256394352461516489831228728, −9.28211316243570547849550038353, −8.23056965752661457611878678442, −7.50106120165131758794553502749, −6.351020105315107701824741824674, −5.54740236126927067787510830984, −3.2844924027633942936411057532, −2.21614119215930781850224066233, −0.55923846131848670844479053371, 1.82294229149937936451754446581, 3.25243538360810220703121480943, 3.824579132981410381955651419417, 5.61383778214724847497904565971, 7.194099393827865576987394450824, 8.19758126607888476007240950921, 9.17863589845911780381357962127, 10.118516348261680653724832136061, 10.608561018696215350759852562134, 11.998066411291001101219696641972, 13.11023115296535885933117921422, 14.113602192632880891496943346324, 15.63133478313935378214846651409, 16.18996838850009549400052352537, 16.948015296099243173817056227529, 18.46949802090251973556228229338, 19.03719615104093059345124653689, 20.13764508164778184455225799634, 20.76073257703401419230124582885, 21.58160534908408890689593320073, 22.55471998456905784794630381473, 23.7298218991970645080525902601, 25.43536506544107322420356219800, 25.73672347289419420575269442607, 26.48931699267126031507537483362

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