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Properties

Label 1-103-103.34-r0-0-0
Degree $1$
Conductor $103$
Sign $-0.727 + 0.686i$
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  + (0.445 + 0.895i)2-s + (−0.850 − 0.526i)3-s + (−0.602 + 0.798i)4-s + (0.739 + 0.673i)5-s + (0.0922 − 0.995i)6-s + (−0.982 + 0.183i)7-s + (−0.982 − 0.183i)8-s + (0.445 + 0.895i)9-s + (−0.273 + 0.961i)10-s + (0.445 + 0.895i)11-s + (0.932 − 0.361i)12-s + (−0.982 + 0.183i)13-s + (−0.602 − 0.798i)14-s + (−0.273 − 0.961i)15-s + (−0.273 − 0.961i)16-s + (0.0922 + 0.995i)17-s + ⋯
L(s)  = 1  + (0.445 + 0.895i)2-s + (−0.850 − 0.526i)3-s + (−0.602 + 0.798i)4-s + (0.739 + 0.673i)5-s + (0.0922 − 0.995i)6-s + (−0.982 + 0.183i)7-s + (−0.982 − 0.183i)8-s + (0.445 + 0.895i)9-s + (−0.273 + 0.961i)10-s + (0.445 + 0.895i)11-s + (0.932 − 0.361i)12-s + (−0.982 + 0.183i)13-s + (−0.602 − 0.798i)14-s + (−0.273 − 0.961i)15-s + (−0.273 − 0.961i)16-s + (0.0922 + 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $-0.727 + 0.686i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ -0.727 + 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2963845134 + 0.7463278600i\)
\(L(\frac12)\) \(\approx\) \(0.2963845134 + 0.7463278600i\)
\(L(1)\) \(\approx\) \(0.6958781527 + 0.5592548323i\)
\(L(1)\) \(\approx\) \(0.6958781527 + 0.5592548323i\)

Euler product

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

−29.27162117190661330866838610821, −28.771035308156636236007237595104, −27.58256088489014868157122569721, −26.79582467898390187526827961810, −25.11944824402792531085422963415, −23.91575896639794796673114921308, −22.912171926541588358439981382432, −21.88930809495031735848432730262, −21.413751620457726900948428557565, −20.136993244733672123395869635705, −19.13379559040842741495902595859, −17.69067279972802592536129347479, −16.79186002814170791581854342459, −15.64070739374718036691207177652, −14.08330196277480616360867711820, −12.99074444195662337292224916925, −12.116060422933513208278602676837, −10.914626297494863306365577260975, −9.754575123135811666132034272313, −9.17224047059701074254625404681, −6.48392084962812385997571659154, −5.46906665267341775053225524493, −4.39704915316434069522495964431, −2.88692929779031452748268340416, −0.7855909640036831593719628381, 2.42149672595080748810533939721, 4.35446602940645023604667125766, 5.876565138730995350057618956619, 6.53193290262339677974729292821, 7.46719700785351584245232018360, 9.34218164281429969265132966167, 10.523844484127511754308712793457, 12.38930704748502972477403164593, 12.80346405775966042252594440537, 14.22867975464113037049929562391, 15.19093682097387504701760242979, 16.70385793648719660775088805587, 17.238117696953590496626265231860, 18.32684277709089870426194017601, 19.33822183070296340198679169271, 21.415718605822215712416810176846, 22.33082021521969475051659250408, 22.795312135125626509163197543988, 23.94398693159310545562728588172, 25.13426550330344606270518871671, 25.67396936269850460589189337739, 26.89965342751493512299858494412, 28.24025963750054290389880209565, 29.371301076885969897560682012223, 30.14430589754033661254695700148

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