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Properties

Label 1-235-235.17-r1-0-0
Degree $1$
Conductor $235$
Sign $0.820 - 0.571i$
Analytic cond. $25.2542$
Root an. cond. $25.2542$
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  + (−0.997 − 0.0682i)2-s + (−0.269 − 0.962i)3-s + (0.990 + 0.136i)4-s + (0.203 + 0.979i)6-s + (−0.730 + 0.682i)7-s + (−0.979 − 0.203i)8-s + (−0.854 + 0.519i)9-s + (−0.917 + 0.398i)11-s + (−0.136 − 0.990i)12-s + (−0.887 − 0.460i)13-s + (0.775 − 0.631i)14-s + (0.962 + 0.269i)16-s + (0.398 − 0.917i)17-s + (0.887 − 0.460i)18-s + (0.576 + 0.816i)19-s + ⋯
L(s)  = 1  + (−0.997 − 0.0682i)2-s + (−0.269 − 0.962i)3-s + (0.990 + 0.136i)4-s + (0.203 + 0.979i)6-s + (−0.730 + 0.682i)7-s + (−0.979 − 0.203i)8-s + (−0.854 + 0.519i)9-s + (−0.917 + 0.398i)11-s + (−0.136 − 0.990i)12-s + (−0.887 − 0.460i)13-s + (0.775 − 0.631i)14-s + (0.962 + 0.269i)16-s + (0.398 − 0.917i)17-s + (0.887 − 0.460i)18-s + (0.576 + 0.816i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $0.820 - 0.571i$
Analytic conductor: \(25.2542\)
Root analytic conductor: \(25.2542\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (1:\ ),\ 0.820 - 0.571i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5825336639 - 0.1830086625i\)
\(L(\frac12)\) \(\approx\) \(0.5825336639 - 0.1830086625i\)
\(L(1)\) \(\approx\) \(0.5068184639 - 0.1203261105i\)
\(L(1)\) \(\approx\) \(0.5068184639 - 0.1203261105i\)

Euler product

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

−26.14577505667243059574426593642, −25.89047785032787545568353649575, −24.22434480822368688230685801065, −23.5544783372782299742502273113, −22.2127217870463526317178611110, −21.417845081768232036829846991097, −20.355994870010013613395133856223, −19.68797715348846472099966115236, −18.64085425527650351468853679956, −17.41730290093057889763132612131, −16.74306073621886966913125046426, −15.97201462471168534802298067680, −15.182681294295643164948746134460, −13.94868295482381026296255080141, −12.43086532926925541454079569669, −11.31092609447975440949444490640, −10.30637756703348794338803917899, −9.850420994160022393423017509369, −8.72944706641227312140578758933, −7.55374149900714214958728568671, −6.39865006410401678900364218437, −5.267210754610773738213066176957, −3.743335809534110342053903287641, −2.57393731614481615184077269426, −0.54708313808260232350787682603, 0.56070487639232543489866845622, 2.18394063248859274069651539236, 2.9389936274030489066461472836, 5.38946448797904859996509869339, 6.282586613619082768132726830575, 7.51187480294470055926380184004, 8.03218495385871187695747700118, 9.49596733476495943912956519961, 10.22874163070565279525578680379, 11.7056941838825340799315577897, 12.2530506074443543738944786661, 13.2290054043265184644936389859, 14.70573115486509418019184203394, 15.88701443277530545508709961984, 16.63677138863795185097816906607, 17.90796882429886192604746772465, 18.29349353036933210333089579495, 19.25574393813484990468542427305, 19.98904147301412932332131024511, 21.07509496783934381971142497238, 22.3975452492293532259545407556, 23.239198448884183636349141653197, 24.57454628293703545475579621847, 24.96544498243454229132837396563, 25.89162693329428754074855746912

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