Properties

Degree $1$
Conductor $103$
Sign $-0.643 - 0.765i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $-0.643 - 0.765i$
Motivic weight: \(0\)
Character: $\chi_{103} (9, \cdot )$
Sato-Tate group: $\mu(17)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ -0.643 - 0.765i)\)

Particular Values

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

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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