Properties

Degree 1
Conductor 373
Sign $0.754 - 0.655i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.0168 + 0.999i)2-s + (0.638 − 0.769i)3-s + (−0.999 + 0.0337i)4-s + (0.283 − 0.959i)5-s + (0.780 + 0.625i)6-s + (0.979 − 0.201i)7-s + (−0.0506 − 0.998i)8-s + (−0.184 − 0.982i)9-s + (0.963 + 0.266i)10-s + (−0.839 − 0.543i)11-s + (−0.612 + 0.790i)12-s + (−0.0506 + 0.998i)13-s + (0.217 + 0.975i)14-s + (−0.557 − 0.830i)15-s + (0.997 − 0.0675i)16-s + (0.528 + 0.848i)17-s + ⋯
L(s,χ)  = 1  + (0.0168 + 0.999i)2-s + (0.638 − 0.769i)3-s + (−0.999 + 0.0337i)4-s + (0.283 − 0.959i)5-s + (0.780 + 0.625i)6-s + (0.979 − 0.201i)7-s + (−0.0506 − 0.998i)8-s + (−0.184 − 0.982i)9-s + (0.963 + 0.266i)10-s + (−0.839 − 0.543i)11-s + (−0.612 + 0.790i)12-s + (−0.0506 + 0.998i)13-s + (0.217 + 0.975i)14-s + (−0.557 − 0.830i)15-s + (0.997 − 0.0675i)16-s + (0.528 + 0.848i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.754 - 0.655i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.754 - 0.655i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(373\)
\( \varepsilon \)  =  $0.754 - 0.655i$
motivic weight  =  \(0\)
character  :  $\chi_{373} (52, \cdot )$
Sato-Tate  :  $\mu(93)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 373,\ (0:\ ),\ 0.754 - 0.655i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.421098884 - 0.5310565324i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.421098884 - 0.5310565324i\)
\(L(\chi,1)\)  \(\approx\)  \(1.259178042 - 0.07953518494i\)
\(L(1,\chi)\)  \(\approx\)  \(1.259178042 - 0.07953518494i\)

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

−25.08828087300528621640822709055, −23.3831828175673037415339590006, −22.781831321329590710803211691877, −21.78302852922426630759099008613, −21.1267672507599536575083687276, −20.56880638316811531091697867269, −19.63222762937777857297707304763, −18.5230635377721230181288734540, −18.048991530102671538644838951555, −16.97980955808027807517782467965, −15.36327981221526098492400862649, −14.84361592868787518932007053011, −13.98950779901840396515110021965, −13.177907003976868140613098262537, −11.83697812081978543731890552131, −10.91315535206767515100254766097, −10.21070258806977779269987323630, −9.589725938899667370946121260233, −8.21848575313999419107261618201, −7.64269642851578883167174325962, −5.538970420349165792280691516041, −4.85141378259319587074502150170, −3.53323486275838622254635890507, −2.73288973100488039548418593360, −1.81982119743230565171238733873, 0.8912058424070358774724098564, 2.14872277990955881260352573360, 3.916693810466954755835244723988, 4.92645998847617967763206610968, 5.91525661804189163178397349604, 7.0496139749481362457666157741, 8.02386805408958039939902379824, 8.6116596818806216595149999559, 9.35221808217869792846832235030, 10.88292482643810543677492391736, 12.38454679667628870723374454082, 13.02217899842226144000177184206, 13.93100917991603258318380802282, 14.52290502254886657251375629351, 15.56461997810154898360263375757, 16.7139965506540698001380020165, 17.29659849945080930197689588965, 18.296479623493454894283073835440, 18.96337131763619707270284199209, 20.11960724621979707616324491514, 21.12258653215973720619931711035, 21.70878407837940392088644160784, 23.54893592763016956501648511446, 23.807999844997373073005893438190, 24.37652544992640320181702682611

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