Properties

Label 1-235-235.43-r0-0-0
Degree $1$
Conductor $235$
Sign $-0.679 - 0.733i$
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  + (0.519 − 0.854i)2-s + (0.816 − 0.576i)3-s + (−0.460 − 0.887i)4-s + (−0.0682 − 0.997i)6-s + (0.269 − 0.962i)7-s + (−0.997 − 0.0682i)8-s + (0.334 − 0.942i)9-s + (0.990 − 0.136i)11-s + (−0.887 − 0.460i)12-s + (−0.631 + 0.775i)13-s + (−0.682 − 0.730i)14-s + (−0.576 + 0.816i)16-s + (−0.136 + 0.990i)17-s + (−0.631 − 0.775i)18-s + (0.203 + 0.979i)19-s + ⋯
L(s)  = 1  + (0.519 − 0.854i)2-s + (0.816 − 0.576i)3-s + (−0.460 − 0.887i)4-s + (−0.0682 − 0.997i)6-s + (0.269 − 0.962i)7-s + (−0.997 − 0.0682i)8-s + (0.334 − 0.942i)9-s + (0.990 − 0.136i)11-s + (−0.887 − 0.460i)12-s + (−0.631 + 0.775i)13-s + (−0.682 − 0.730i)14-s + (−0.576 + 0.816i)16-s + (−0.136 + 0.990i)17-s + (−0.631 − 0.775i)18-s + (0.203 + 0.979i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $-0.679 - 0.733i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ -0.679 - 0.733i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7721786007 - 1.767674546i\)
\(L(\frac12)\) \(\approx\) \(0.7721786007 - 1.767674546i\)
\(L(1)\) \(\approx\) \(1.160859463 - 1.161284950i\)
\(L(1)\) \(\approx\) \(1.160859463 - 1.161284950i\)

Euler product

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

−26.41966709637619268631681709898, −25.486052780398103983003866390511, −24.8136896717868707797068909225, −24.26734611965043621618012241992, −22.61061834408456030120738696761, −22.13815588064915183171072853287, −21.2807164467756670889936775104, −20.26910153629318332517784038733, −19.22257451042108391036355639585, −17.953820872566554027473666075, −17.08700801977752605602258886737, −15.72271907820046682213527918949, −15.38991881087232173809200952884, −14.38340112917488107211969335674, −13.65768989952520846031042600486, −12.428769468304993813635245731908, −11.44752469466657980603463644992, −9.64509344032772079235434220567, −9.00520521200575641922527020765, −7.98531266238643894625167442965, −6.975509126937157966072033410288, −5.49666063204523709648098452441, −4.6992519579089732325853911768, −3.433110266325887561304626433, −2.39452671617256007742223821574, 1.23506944913173743332374002388, 2.18176108424521934866306113088, 3.72211447453510202468229673674, 4.253557114913464075765378792121, 6.07352727485997616152812030721, 7.131351557716418485965908108813, 8.44419414342818798385796282759, 9.496605051283705138970655602455, 10.46491986927122377120770359238, 11.725141804958790845693156588346, 12.53148809940903995020387767820, 13.571176343244735273952369910785, 14.37901922991520454847567844189, 14.817450788802309291627672818933, 16.61731528542766520009534315076, 17.72113187567037354522407990043, 18.88731625185263425704367620436, 19.53353136892897646401174141792, 20.2998986813160756661253265869, 21.06985668684930297926143806657, 22.143543636066256152209134582475, 23.162491905985189844199981315014, 24.19044468524160338592400832211, 24.55951068015821711182244843090, 26.118192454496738492248893985950

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