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Properties

Label 1-235-235.113-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: $\chi_{235} (113, \cdot )$
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

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

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