Properties

Degree 1
Conductor 137
Sign $0.796 + 0.605i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.796 + 0.605i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.796 + 0.605i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(137\)
\( \varepsilon \)  =  $0.796 + 0.605i$
motivic weight  =  \(0\)
character  :  $\chi_{137} (25, \cdot )$
Sato-Tate  :  $\mu(68)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 137,\ (0:\ ),\ 0.796 + 0.605i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.071981741 + 0.6979332539i$
$L(\frac12,\chi)$  $\approx$  $2.071981741 + 0.6979332539i$
$L(\chi,1)$  $\approx$  1.938123596 + 0.4600708015i
$L(1,\chi)$  $\approx$  1.938123596 + 0.4600708015i

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

−29.06864859993452510478758119830, −27.15735737654488019804795270963, −26.02248929789274367992568106758, −25.54637371236213754154203438288, −24.133793098801661194694435894709, −23.39433694713962745226452965894, −22.640685777619733452157026508378, −21.29090795210259636372293561766, −20.455983104865591902252876209064, −19.30034973713781154001924210553, −18.85598970143517089444645868371, −17.191592476230476331480548413511, −15.70554583535751168033078773401, −14.69405734116010725205512914820, −14.062846999214036784489356750390, −13.02305706762173621850831030994, −12.122943322239688933845540743604, −10.66260532059741564553612316132, −9.80783807648440688894157847199, −7.49959611532142168771808468512, −7.287458943675306818583675995534, −5.854312086441384067468397087246, −3.994516722410478190541534835280, −3.12218093991697948630942403005, −1.92063797141014760484742385133, 2.3895991826787546964786722347, 3.32900667865979175528573510888, 4.904222240161818730156943042470, 5.41008499575876349893528744419, 7.37696955286965869655562964341, 8.453458137893226119175301315489, 9.555098043319206517615770329532, 11.00210114104570870151101846988, 12.5032681232611110574762974144, 13.013569501286454308541031047180, 14.306082586628134718291580752892, 15.37000248714374164571833578024, 15.97275569429190009312851620290, 16.90155120432124089935330157735, 18.85756364187766887172505745758, 19.94943695446875469356973944903, 20.73527451141350575977061261449, 21.57601936317813420269623896108, 22.394452511532344347852880608031, 23.764544940837951756179267564072, 24.69288065191875783916047405799, 25.35506236302514824374457817884, 26.352832057466981702655257581170, 27.65976841962654813177353282574, 28.651182226751401304524851619360

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