Properties

Degree 1
Conductor 137
Sign $0.358 + 0.933i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.602 − 0.798i)2-s + (−0.995 + 0.0922i)3-s + (−0.273 − 0.961i)4-s + (−0.798 + 0.602i)5-s + (−0.526 + 0.850i)6-s + (−0.739 + 0.673i)7-s + (−0.932 − 0.361i)8-s + (0.982 − 0.183i)9-s + i·10-s + (0.273 + 0.961i)11-s + (0.361 + 0.932i)12-s + (0.673 + 0.739i)13-s + (0.0922 + 0.995i)14-s + (0.739 − 0.673i)15-s + (−0.850 + 0.526i)16-s + (−0.932 + 0.361i)17-s + ⋯
L(s,χ)  = 1  + (0.602 − 0.798i)2-s + (−0.995 + 0.0922i)3-s + (−0.273 − 0.961i)4-s + (−0.798 + 0.602i)5-s + (−0.526 + 0.850i)6-s + (−0.739 + 0.673i)7-s + (−0.932 − 0.361i)8-s + (0.982 − 0.183i)9-s + i·10-s + (0.273 + 0.961i)11-s + (0.361 + 0.932i)12-s + (0.673 + 0.739i)13-s + (0.0922 + 0.995i)14-s + (0.739 − 0.673i)15-s + (−0.850 + 0.526i)16-s + (−0.932 + 0.361i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.358 + 0.933i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.358 + 0.933i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(137\)
\( \varepsilon \)  =  $0.358 + 0.933i$
motivic weight  =  \(0\)
character  :  $\chi_{137} (76, \cdot )$
Sato-Tate  :  $\mu(68)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 137,\ (0:\ ),\ 0.358 + 0.933i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4242425554 + 0.2916241824i$
$L(\frac12,\chi)$  $\approx$  $0.4242425554 + 0.2916241824i$
$L(\chi,1)$  $\approx$  0.7048506051 - 0.04284088489i
$L(1,\chi)$  $\approx$  0.7048506051 - 0.04284088489i

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

−28.18428861449588951228488230851, −27.22561081298159767102547210709, −26.38755092369798962012272811426, −25.00867456442623614420617386171, −24.017976952755076869199857481936, −23.45200830967322377362000918485, −22.591777386481596206940330362940, −21.74567775204308385006298585037, −20.39363828033587003829939788550, −19.18190088755325889338959405550, −17.74771853400212538176905472410, −16.88662980780234201766550367901, −15.97374160285453629186391514404, −15.52126673581861956715684539538, −13.54788121966026086997562961095, −13.02387762549547079365622978983, −11.81511531051042141918407490366, −10.89972878576891900993371483682, −9.08675289732796942571888875743, −7.80076178547607758221486442749, −6.71531642494221386851893631791, −5.73374167341099945615655500543, −4.4674718471627861818645030031, −3.51506795882738142372721228490, −0.42299776013398737073278214221, 1.95208307793027773408796043356, 3.65837089157144504964038264602, 4.56208812474426064222535966950, 6.11280942599868303922035546851, 6.82677018250908939633822911470, 8.984575125789860833017603382778, 10.29865895688140272544532370303, 11.07706724138858603281906968430, 12.23215897392189387878337268878, 12.60518495398618981892303276559, 14.32189945605433009663975124744, 15.4264973688724018505816387805, 16.173937857315136259065481770324, 17.90618561170097214885065775350, 18.718004049720838848649684471336, 19.55237304587397121256309672428, 20.83475652315625555086121907580, 22.07587390412846905105769810065, 22.53449773637129063257243602151, 23.354382570458922302868580127539, 24.21573245764510892137338414581, 25.7794120764310943897320322746, 27.13374639691748415005385396627, 28.00091408007022793360073019581, 28.681747204268212120499712434916

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