Properties

Degree 1
Conductor 137
Sign $0.995 - 0.0946i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.183 + 0.982i)2-s + (0.990 + 0.138i)3-s + (−0.932 + 0.361i)4-s + (−0.565 − 0.824i)5-s + (0.0461 + 0.998i)6-s + (0.895 − 0.445i)7-s + (−0.526 − 0.850i)8-s + (0.961 + 0.273i)9-s + (0.707 − 0.707i)10-s + (−0.361 − 0.932i)11-s + (−0.973 + 0.228i)12-s + (−0.948 − 0.317i)13-s + (0.602 + 0.798i)14-s + (−0.445 − 0.895i)15-s + (0.739 − 0.673i)16-s + (0.526 − 0.850i)17-s + ⋯
L(s,χ)  = 1  + (0.183 + 0.982i)2-s + (0.990 + 0.138i)3-s + (−0.932 + 0.361i)4-s + (−0.565 − 0.824i)5-s + (0.0461 + 0.998i)6-s + (0.895 − 0.445i)7-s + (−0.526 − 0.850i)8-s + (0.961 + 0.273i)9-s + (0.707 − 0.707i)10-s + (−0.361 − 0.932i)11-s + (−0.973 + 0.228i)12-s + (−0.948 − 0.317i)13-s + (0.602 + 0.798i)14-s + (−0.445 − 0.895i)15-s + (0.739 − 0.673i)16-s + (0.526 − 0.850i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.995 - 0.0946i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.995 - 0.0946i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(137\)
\( \varepsilon \)  =  $0.995 - 0.0946i$
motivic weight  =  \(0\)
character  :  $\chi_{137} (27, \cdot )$
Sato-Tate  :  $\mu(136)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 137,\ (1:\ ),\ 0.995 - 0.0946i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(2.255515177 - 0.1069750651i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(2.255515177 - 0.1069750651i\)
\(L(\chi,1)\)  \(\approx\)  \(1.423695633 + 0.2674746263i\)
\(L(1,\chi)\)  \(\approx\)  \(1.423695633 + 0.2674746263i\)

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

−28.21016799559550325809240245689, −27.19705335731837261515199999013, −26.58810020642108886578649947399, −25.43293314081999413772106193382, −24.087153549197679673138004608330, −23.26520625573797733511850983866, −21.898763269763015983903338471312, −21.26412064104619453519879596126, −20.060653970212361305745769248383, −19.43636446913797046075981080357, −18.40219226784378330035557656084, −17.70220038526156883689739198732, −15.43362442808468399832236713713, −14.68221888082266263834308722446, −14.010462036342544658420164854787, −12.53956137337950786704744108681, −11.76520701837270970113259692574, −10.447711245514993210172618038751, −9.49938465036715914879401086566, −8.17424616547121112393897247353, −7.26721523664236787950369015728, −5.112039196588245316929417266714, −3.8757798255797715141270423687, −2.705313463063759891196736414079, −1.69212003489164449376568209492, 0.80126143295763003344151986829, 3.12799411488601692643792703872, 4.46474190312774140584453642235, 5.25250340530594647992031001639, 7.31201694165306180608610639587, 7.98928182527259666895531764225, 8.792516703255209254925773732998, 10.025178176082439828570699672057, 11.87175071049904444921729496234, 13.12319103733660980679536552077, 14.04967312547905335790508865497, 14.84605951499368267070429616118, 16.0001620720277396702522919292, 16.66331805274786421216007863328, 18.033608440491744799032126980604, 19.124880110529201295917192003992, 20.35746991772198163967708684556, 21.092015855071583699625761568657, 22.33259308272645848911912692542, 23.74004584855442444349154667504, 24.407495975093467015558179695274, 24.94353212331000751140686225147, 26.33113848351065715469893107544, 27.146661676284079602903642365871, 27.51555478631591880674061037276

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