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Properties

Label 1-235-235.52-r0-0-0
Degree $1$
Conductor $235$
Sign $0.908 + 0.416i$
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  + (−0.631 − 0.775i)2-s + (0.398 + 0.917i)3-s + (−0.203 + 0.979i)4-s + (0.460 − 0.887i)6-s + (0.942 − 0.334i)7-s + (0.887 − 0.460i)8-s + (−0.682 + 0.730i)9-s + (0.576 + 0.816i)11-s + (−0.979 + 0.203i)12-s + (0.997 − 0.0682i)13-s + (−0.854 − 0.519i)14-s + (−0.917 − 0.398i)16-s + (−0.816 − 0.576i)17-s + (0.997 + 0.0682i)18-s + (−0.990 + 0.136i)19-s + ⋯
L(s)  = 1  + (−0.631 − 0.775i)2-s + (0.398 + 0.917i)3-s + (−0.203 + 0.979i)4-s + (0.460 − 0.887i)6-s + (0.942 − 0.334i)7-s + (0.887 − 0.460i)8-s + (−0.682 + 0.730i)9-s + (0.576 + 0.816i)11-s + (−0.979 + 0.203i)12-s + (0.997 − 0.0682i)13-s + (−0.854 − 0.519i)14-s + (−0.917 − 0.398i)16-s + (−0.816 − 0.576i)17-s + (0.997 + 0.0682i)18-s + (−0.990 + 0.136i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $0.908 + 0.416i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ 0.908 + 0.416i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.059790425 + 0.2314496761i\)
\(L(\frac12)\) \(\approx\) \(1.059790425 + 0.2314496761i\)
\(L(1)\) \(\approx\) \(0.9712275889 + 0.05324801297i\)
\(L(1)\) \(\approx\) \(0.9712275889 + 0.05324801297i\)

Euler product

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

−26.061059789918706281187641154852, −25.18487973159536681112523630108, −24.46890189765371248817872847690, −23.82983641742479199636696772478, −22.97165365066890100616437615596, −21.48046936788372556626995699324, −20.384030671380947501613251603180, −19.18519274023406701830763413955, −18.83362016911096461113438256634, −17.57833876941956533972424481388, −17.24178708509566623356717452818, −15.709196757868675796749842120051, −14.86416001717342873423414457795, −13.95110336979702853332281983791, −13.1739630248555929501218958863, −11.57580080675248193371995593633, −10.85378361409642320292678930711, −9.04961591960775688453141078939, −8.56277200394888358743153297362, −7.683776828070529041609942309708, −6.44644559546557265357174086085, −5.779087876001458754750505729186, −4.120918234301886789105741099485, −2.194011597018744106275781168336, −1.091357225533516872647406225487, 1.52143800125103087630447889137, 2.79486071755484931317502438300, 4.14996851990273401298570900013, 4.74193630240136873172039819929, 6.81509474531283355070812607618, 8.22979625251510212498154576300, 8.81096147490367099853423518952, 9.91407986893475531849987600612, 10.86959257856539358353981200348, 11.45445582184427132680128873517, 12.8617711930474707298177968000, 13.990645792279394378037494090621, 14.96142969553482837706046931782, 16.106734759406609456085426673857, 17.104345557729995437211893730502, 17.84966539296836356812693232709, 18.99460624856402983349736761721, 20.18582619116956332930306838475, 20.55547361691260297539508010200, 21.395980256737836177982158104028, 22.34715281727081212408477011593, 23.27461220734271383037136934144, 24.919784745823605663690483826216, 25.63666841182306174898624693036, 26.61781229727419031093860552733

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