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Properties

Label 1-103-103.66-r0-0-0
Degree $1$
Conductor $103$
Sign $-0.0419 + 0.999i$
Analytic cond. $0.478329$
Root an. cond. $0.478329$
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  + (0.739 + 0.673i)2-s + (0.932 + 0.361i)3-s + (0.0922 + 0.995i)4-s + (−0.850 + 0.526i)5-s + (0.445 + 0.895i)6-s + (−0.602 − 0.798i)7-s + (−0.602 + 0.798i)8-s + (0.739 + 0.673i)9-s + (−0.982 − 0.183i)10-s + (0.739 + 0.673i)11-s + (−0.273 + 0.961i)12-s + (−0.602 − 0.798i)13-s + (0.0922 − 0.995i)14-s + (−0.982 + 0.183i)15-s + (−0.982 + 0.183i)16-s + (0.445 − 0.895i)17-s + ⋯
L(s)  = 1  + (0.739 + 0.673i)2-s + (0.932 + 0.361i)3-s + (0.0922 + 0.995i)4-s + (−0.850 + 0.526i)5-s + (0.445 + 0.895i)6-s + (−0.602 − 0.798i)7-s + (−0.602 + 0.798i)8-s + (0.739 + 0.673i)9-s + (−0.982 − 0.183i)10-s + (0.739 + 0.673i)11-s + (−0.273 + 0.961i)12-s + (−0.602 − 0.798i)13-s + (0.0922 − 0.995i)14-s + (−0.982 + 0.183i)15-s + (−0.982 + 0.183i)16-s + (0.445 − 0.895i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0419 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0419 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $-0.0419 + 0.999i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (66, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ -0.0419 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.140802978 + 1.189696852i\)
\(L(\frac12)\) \(\approx\) \(1.140802978 + 1.189696852i\)
\(L(1)\) \(\approx\) \(1.360066744 + 0.8717924032i\)
\(L(1)\) \(\approx\) \(1.360066744 + 0.8717924032i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (0.739 + 0.673i)T \)
3 \( 1 + (0.932 + 0.361i)T \)
5 \( 1 + (-0.850 + 0.526i)T \)
7 \( 1 + (-0.602 - 0.798i)T \)
11 \( 1 + (0.739 + 0.673i)T \)
13 \( 1 + (-0.602 - 0.798i)T \)
17 \( 1 + (0.445 - 0.895i)T \)
19 \( 1 + (0.932 - 0.361i)T \)
23 \( 1 + (0.739 - 0.673i)T \)
29 \( 1 + (-0.850 + 0.526i)T \)
31 \( 1 + (-0.982 + 0.183i)T \)
37 \( 1 + (-0.273 + 0.961i)T \)
41 \( 1 + (-0.850 - 0.526i)T \)
43 \( 1 + (-0.273 - 0.961i)T \)
47 \( 1 + T \)
53 \( 1 + (0.932 - 0.361i)T \)
59 \( 1 + (-0.602 + 0.798i)T \)
61 \( 1 + (0.445 - 0.895i)T \)
67 \( 1 + (-0.602 + 0.798i)T \)
71 \( 1 + (-0.850 - 0.526i)T \)
73 \( 1 + (-0.850 - 0.526i)T \)
79 \( 1 + (-0.850 + 0.526i)T \)
83 \( 1 + (-0.602 - 0.798i)T \)
89 \( 1 + (0.0922 - 0.995i)T \)
97 \( 1 + (0.445 + 0.895i)T \)
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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

−29.70100978530618981190354672932, −28.73017992733886613433517064063, −27.63624660847634166558859630093, −26.55746599709175935224825864841, −25.004203741445844838310630475415, −24.36472021496212213056940215748, −23.4073568375505289719573322447, −22.06531021986202076646995927138, −21.17723128191632566881864854038, −19.961695452905480351987654936506, −19.283682303748319716398990249256, −18.6929150957765598343466984170, −16.47879327379581488143909477864, −15.290355633516756591477428054497, −14.48121101762412024271083626350, −13.228907141936707428645636775593, −12.31018613914307323330603560411, −11.521683432473167706630589486293, −9.58960303429168735579443538631, −8.82721984011645036442618390280, −7.22612539074699924418348833165, −5.72114768818212864072680181299, −4.0144558378376591887840390920, −3.16370987827969657185360309463, −1.55657860076979911948408245626, 2.94104912582558894217159137118, 3.72761094299364522468608076907, 4.93060930895517019847512958055, 7.16296208881469373879780420517, 7.35486829946460677828287205912, 8.97111980141245340849749766808, 10.34820917193139078478897277420, 11.93894859249209497725872080911, 13.134200907261336809116952677094, 14.26716680710627832940176314343, 15.0096166668954899316632027329, 15.93812753222040188143229578920, 16.95872957774105138356258634806, 18.59310572458332865517451389436, 20.0716426711943265052176054935, 20.38146738645526125089310009212, 22.249855559550315676669003838348, 22.59480542376526760296506364634, 23.86771445931088540899334772527, 25.0336274432218032866396111860, 25.83478013403104345443589375524, 26.85397847527864224451272511079, 27.3861461862216298579237236505, 29.55595424066230905324625910838, 30.454894206285918156590113937204

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