Properties

Degree 1
Conductor 103
Sign $-0.635 + 0.772i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.982 − 0.183i)2-s + (0.0922 − 0.995i)3-s + (0.932 + 0.361i)4-s + (−0.602 + 0.798i)5-s + (−0.273 + 0.961i)6-s + (−0.850 + 0.526i)7-s + (−0.850 − 0.526i)8-s + (−0.982 − 0.183i)9-s + (0.739 − 0.673i)10-s + (−0.982 − 0.183i)11-s + (0.445 − 0.895i)12-s + (−0.850 + 0.526i)13-s + (0.932 − 0.361i)14-s + (0.739 + 0.673i)15-s + (0.739 + 0.673i)16-s + (−0.273 − 0.961i)17-s + ⋯
L(s,χ)  = 1  + (−0.982 − 0.183i)2-s + (0.0922 − 0.995i)3-s + (0.932 + 0.361i)4-s + (−0.602 + 0.798i)5-s + (−0.273 + 0.961i)6-s + (−0.850 + 0.526i)7-s + (−0.850 − 0.526i)8-s + (−0.982 − 0.183i)9-s + (0.739 − 0.673i)10-s + (−0.982 − 0.183i)11-s + (0.445 − 0.895i)12-s + (−0.850 + 0.526i)13-s + (0.932 − 0.361i)14-s + (0.739 + 0.673i)15-s + (0.739 + 0.673i)16-s + (−0.273 − 0.961i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.635 + 0.772i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.635 + 0.772i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(103\)
\( \varepsilon \)  =  $-0.635 + 0.772i$
motivic weight  =  \(0\)
character  :  $\chi_{103} (61, \cdot )$
Sato-Tate  :  $\mu(17)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 103,\ (0:\ ),\ -0.635 + 0.772i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.04168586749 + 0.08831236191i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.04168586749 + 0.08831236191i\)
\(L(\chi,1)\)  \(\approx\)  \(0.3956082431 - 0.04221026282i\)
\(L(1,\chi)\)  \(\approx\)  \(0.3956082431 - 0.04221026282i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−28.79515114993771375205733250574, −28.384103952946855011935317686363, −27.30969960167479459161967663875, −26.39787811694380776053858607865, −25.802957665845081144242324880638, −24.36547089852374082470897041572, −23.39995281616782247758717882981, −22.03136457624947501477039399927, −20.62155529091431065642641420804, −20.02527235255970967601649745219, −19.18441148184216861860395201730, −17.43557488177031592857358584118, −16.69602162658274850776173295665, −15.68937400707759000039129761022, −15.14953388992378808251301332445, −13.191052103297639661166017510270, −11.77511628386686976363214637457, −10.43646512948771703979794706493, −9.743018809559946509650734765723, −8.52502011551783174997826419666, −7.53132359200058724996302202763, −5.78814977501620317344454431332, −4.35425752090854493130775656389, −2.74873890944640246571219922075, −0.11937630649516772142456639082, 2.263851458798049051365397114129, 3.1670928688305020161793501191, 5.99565969440842758313248500878, 7.127638981116794683566840181288, 7.86907643282660414799633970357, 9.234504360657451724353003801402, 10.53535408074579628349904476255, 11.82482064178857669342024907715, 12.50236718768801695663604402895, 14.12044015719124581988651498071, 15.51705456393430623935485779238, 16.47327122194201560668126132968, 18.01099621548817158434219068878, 18.63494871616104815616238106326, 19.33472516657565864651053615920, 20.27392660022116367310572143747, 21.85717286693382343302739750708, 23.03687500734968186706548732863, 24.19524507322608961091443952819, 25.231486090121214348292615363131, 26.14049935308555607161945472736, 26.889192959839967401000526646205, 28.30291050889813599163385961578, 29.25357212713725462532865405311, 29.765830730192881624817219244805

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