↑

Properties

Label 1-235-235.117-r0-0-0
Degree $1$
Conductor $235$
Sign $0.987 + 0.155i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.269 + 0.962i)2-s + (0.887 − 0.460i)3-s + (−0.854 + 0.519i)4-s + (0.682 + 0.730i)6-s + (−0.136 − 0.990i)7-s + (−0.730 − 0.682i)8-s + (0.576 − 0.816i)9-s + (0.0682 − 0.997i)11-s + (−0.519 + 0.854i)12-s + (0.942 − 0.334i)13-s + (0.917 − 0.398i)14-s + (0.460 − 0.887i)16-s + (0.997 − 0.0682i)17-s + (0.942 + 0.334i)18-s + (−0.775 + 0.631i)19-s + ⋯
L(s)  = 1  + (0.269 + 0.962i)2-s + (0.887 − 0.460i)3-s + (−0.854 + 0.519i)4-s + (0.682 + 0.730i)6-s + (−0.136 − 0.990i)7-s + (−0.730 − 0.682i)8-s + (0.576 − 0.816i)9-s + (0.0682 − 0.997i)11-s + (−0.519 + 0.854i)12-s + (0.942 − 0.334i)13-s + (0.917 − 0.398i)14-s + (0.460 − 0.887i)16-s + (0.997 − 0.0682i)17-s + (0.942 + 0.334i)18-s + (−0.775 + 0.631i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $0.987 + 0.155i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ 0.987 + 0.155i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.662316840 + 0.1302128051i\)
\(L(\frac12)\) \(\approx\) \(1.662316840 + 0.1302128051i\)
\(L(1)\) \(\approx\) \(1.422528195 + 0.2374067629i\)
\(L(1)\) \(\approx\) \(1.422528195 + 0.2374067629i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−26.133804548246408311444902743171, −25.591411416127770819626006673993, −24.41917712826383648446291692133, −23.21660440169012019315840325029, −22.26678703667359244712216984652, −21.411899752574441601969518653093, −20.73193954544170358212829188145, −19.930490594268635742602203012370, −18.81953053695600593511192706578, −18.41333643345700753967071203093, −16.840246984714072452021097297561, −15.35277973477819027163843891313, −14.901929094050236955551387870545, −13.76296393810648638909406161193, −12.84276430988350437585895913566, −11.92546016484065093294648720885, −10.70909493488546434285542301344, −9.70619971998014176721381400278, −8.98318500715823914418735769669, −8.0184064163856086212101373108, −6.17311370562495313455952425067, −4.826864011051648742173232329494, −3.88915485707489211941471780971, −2.68892061199989861618208294474, −1.81934560688126725731441117028, 1.163730666738803991653089634062, 3.36903047542206493584227706302, 3.83599825009819170374567496426, 5.59300736861731764606685309011, 6.64025973781519920754415034991, 7.66075480547801267942784735605, 8.37096947031743249855081246272, 9.406014943257605601582651612904, 10.70310010799822586432532490913, 12.387056053510128066365499226263, 13.30696976355603485572715058580, 13.984014460887177731361707266319, 14.69134963715312263440579245806, 15.944942599934130777583481828459, 16.655717346224644001409772285, 17.84868484462015328236423387457, 18.72925388740282549073798611607, 19.650814572341511734543813292053, 20.848356457975302061477319426736, 21.58978822451033427494191659071, 23.11485949075224715064482562925, 23.56938932904671819361704680534, 24.466178349737330253231117341300, 25.551660783805425253108662126607, 25.8914045891226434713596000471

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