Properties

Degree 1
Conductor 113
Sign $-0.940 - 0.340i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.623 + 0.781i)2-s + (0.330 + 0.943i)3-s + (−0.222 − 0.974i)4-s + (−0.330 + 0.943i)5-s + (−0.943 − 0.330i)6-s + (−0.900 − 0.433i)7-s + (0.900 + 0.433i)8-s + (−0.781 + 0.623i)9-s + (−0.532 − 0.846i)10-s + (−0.974 − 0.222i)11-s + (0.846 − 0.532i)12-s + (−0.433 + 0.900i)13-s + (0.900 − 0.433i)14-s − 15-s + (−0.900 + 0.433i)16-s + (0.993 − 0.111i)17-s + ⋯
L(s,χ)  = 1  + (−0.623 + 0.781i)2-s + (0.330 + 0.943i)3-s + (−0.222 − 0.974i)4-s + (−0.330 + 0.943i)5-s + (−0.943 − 0.330i)6-s + (−0.900 − 0.433i)7-s + (0.900 + 0.433i)8-s + (−0.781 + 0.623i)9-s + (−0.532 − 0.846i)10-s + (−0.974 − 0.222i)11-s + (0.846 − 0.532i)12-s + (−0.433 + 0.900i)13-s + (0.900 − 0.433i)14-s − 15-s + (−0.900 + 0.433i)16-s + (0.993 − 0.111i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.940 - 0.340i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.940 - 0.340i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(113\)
\( \varepsilon \)  =  $-0.940 - 0.340i$
motivic weight  =  \(0\)
character  :  $\chi_{113} (13, \cdot )$
Sato-Tate  :  $\mu(56)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 113,\ (0:\ ),\ -0.940 - 0.340i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.08028866876 + 0.4576198386i$
$L(\frac12,\chi)$  $\approx$  $-0.08028866876 + 0.4576198386i$
$L(\chi,1)$  $\approx$  0.3804526868 + 0.4693711288i
$L(1,\chi)$  $\approx$  0.3804526868 + 0.4693711288i

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

−28.75633647808961094086169514042, −28.16164435110967719188007609581, −26.863601497565526143811820692530, −25.5633879449187219389654906868, −25.133011076431885759148147150323, −23.667435196907458976543935520552, −22.717260539352723101741141569594, −21.17843126947092920330584660713, −20.31110085910757421169473862956, −19.38404423531395210768406366218, −18.72784160439452943967414816437, −17.52701041464412620720479633183, −16.54789502340291474410593887625, −15.20743633897383715828806751208, −13.21070349518407330842711983993, −12.75985157816939352795590096415, −12.00242238418008882636925843596, −10.37982048970252845088690306189, −9.09702206469322149952504345931, −8.22534299923429012024920666837, −7.2048829228124353636924167368, −5.35652427657658312905815843115, −3.412016537131324187598971561150, −2.25238369765044463702786916785, −0.50641649341030994244022848258, 2.75499895429718495642106622042, 4.18853819713499077118240535878, 5.73662050611423354350398440177, 7.03608671510722730328175363088, 8.09580449431634128814451844978, 9.544010976341065159071731230343, 10.23998537260842198011506048657, 11.21901600524772936384654938648, 13.42142778175823003479785884498, 14.55279185354559743535036678262, 15.29617565963200803635910596156, 16.32930300611126660588859366235, 17.0443549285958486630861808916, 18.85458252228046909462839700597, 19.13862674826660623750125761982, 20.545115916781598738593912620959, 21.87239852115335617973548508256, 22.96763141470526956015855238411, 23.640186710662394640036758790053, 25.395748003959103934455993818806, 26.05246881228614713616017855433, 26.66714900667587212397385102546, 27.49743277459007830038260345072, 28.68242712011277866668025083862, 29.689699165890558454925392278219

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