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Properties

Label 1-103-103.23-r0-0-0
Degree $1$
Conductor $103$
Sign $-0.643 + 0.765i$
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  + (−0.602 + 0.798i)2-s + (0.445 + 0.895i)3-s + (−0.273 − 0.961i)4-s + (0.0922 + 0.995i)5-s + (−0.982 − 0.183i)6-s + (0.932 − 0.361i)7-s + (0.932 + 0.361i)8-s + (−0.602 + 0.798i)9-s + (−0.850 − 0.526i)10-s + (−0.602 + 0.798i)11-s + (0.739 − 0.673i)12-s + (0.932 − 0.361i)13-s + (−0.273 + 0.961i)14-s + (−0.850 + 0.526i)15-s + (−0.850 + 0.526i)16-s + (−0.982 + 0.183i)17-s + ⋯
L(s)  = 1  + (−0.602 + 0.798i)2-s + (0.445 + 0.895i)3-s + (−0.273 − 0.961i)4-s + (0.0922 + 0.995i)5-s + (−0.982 − 0.183i)6-s + (0.932 − 0.361i)7-s + (0.932 + 0.361i)8-s + (−0.602 + 0.798i)9-s + (−0.850 − 0.526i)10-s + (−0.602 + 0.798i)11-s + (0.739 − 0.673i)12-s + (0.932 − 0.361i)13-s + (−0.273 + 0.961i)14-s + (−0.850 + 0.526i)15-s + (−0.850 + 0.526i)16-s + (−0.982 + 0.183i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $-0.643 + 0.765i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ -0.643 + 0.765i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3755029153 + 0.8065135169i\)
\(L(\frac12)\) \(\approx\) \(0.3755029153 + 0.8065135169i\)
\(L(1)\) \(\approx\) \(0.6651794975 + 0.6332341286i\)
\(L(1)\) \(\approx\) \(0.6651794975 + 0.6332341286i\)

Euler product

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

−29.2277732644412525232681541064, −28.644096490158395271793497026859, −27.54608863628002735379759269624, −26.446900343929606059918948784143, −25.28892015347564615227582503132, −24.410127045667098770595061882672, −23.47141240028799331144929662654, −21.672948282529068464298194136029, −20.74233794236017493805961825145, −20.12057971260340650096557469388, −18.78983907485672880688798745875, −18.136123423148221626192156938684, −17.09321481165849499513489312690, −15.79591542399222767530465511898, −13.85256245041753597783962553827, −13.20419618517115270994945104625, −11.972974659509039570215049402120, −11.19521239261523937383133115179, −9.34862012039191773625259126671, −8.42055761806393024718582234388, −7.79570079974489125268165727373, −5.77260155068789358726301252115, −3.9995757657824352128288993579, −2.27902694535024426453932567501, −1.17567859798919861865275415529, 2.25608619753771524378677105248, 4.17934145067616032954499344198, 5.41753733351820716984887740039, 7.00727779936844812388516938691, 8.0726075412503567934519134541, 9.21186459406482860489432196288, 10.63756495955161224637008323145, 10.88502073324069292696810158488, 13.57103374265704863193968732662, 14.514500965154560834323602092147, 15.27687174352026059634782121761, 16.15129113081687937469501771564, 17.721238463641696454040030543187, 18.15907450106818350915221475794, 19.74623236415982044765882612512, 20.5943542958242903862499638578, 21.98857026778235795930744483098, 22.97999778788272461197115285886, 24.08055318821638468879424237517, 25.474100102771091175579806889898, 26.12984899390833832631132418671, 26.87548838717958326612480338748, 27.7773240555771884594830272636, 28.721017363306198769446646020893, 30.51831961841773245389956608594

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