Properties

Degree 1
Conductor 137
Sign $0.627 + 0.778i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.627 + 0.778i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.627 + 0.778i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(137\)
\( \varepsilon \)  =  $0.627 + 0.778i$
motivic weight  =  \(0\)
character  :  $\chi_{137} (30, \cdot )$
Sato-Tate  :  $\mu(68)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 137,\ (0:\ ),\ 0.627 + 0.778i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4563953361 + 0.2183608483i$
$L(\frac12,\chi)$  $\approx$  $0.4563953361 + 0.2183608483i$
$L(\chi,1)$  $\approx$  0.5488531997 + 0.1487829156i
$L(1,\chi)$  $\approx$  0.5488531997 + 0.1487829156i

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

−28.37702710047051894026775258363, −27.52539005596494234885944014935, −26.74125659577943519083802802202, −25.844637201766647716190052755040, −24.1870851116917991687508502447, −22.94864557741138510903402734531, −22.313911610774186877060514108227, −21.27723696475020720107390294437, −20.38776236797571092928992932985, −19.20715831822434657907164922273, −18.603862783580846230311980195952, −16.83845638969703190593294632839, −16.529776915799219671768772087068, −15.40345980384933245836681489816, −13.68096844881933839304026306386, −12.248500005157836287864807074744, −11.72524514114540895091895207769, −10.72906913074986688106239286528, −9.34658188576074907391994934894, −8.84705942607752935589172684860, −7.0787518063209098183819606215, −5.41392913885446737474277854183, −3.99890438960547748399822385994, −3.25130761931502418441548848487, −0.79976306551900521127457776615, 1.04762329362326425832682218922, 3.5207691275327260553537352736, 5.19568856396930713848273810571, 6.44943181181739642936019968576, 7.24898624964629256484492716255, 8.05802479070355343236696749092, 9.755315246965305760870474567802, 10.76627276469201228066679986039, 12.13570278757167209650809623476, 13.16651385396602869590970013688, 14.45299467461571416237940704168, 15.59219668068290060670556842716, 16.49730737882016278426019955341, 17.41691079394467259675782934520, 18.49329475279832580266973463905, 19.23616350115385939924289948514, 20.08884765760548098497901614653, 22.45593145932335597032991132900, 22.92147076928922488773240073423, 23.46753309418670447814044129216, 24.81324959370033653396039538602, 25.47879982970744194158542745888, 26.63263969881810372209039621631, 27.66811609428687515446127596315, 28.326464936698842037800100942790

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