Properties

Degree 1
Conductor 137
Sign $0.449 + 0.893i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.273 + 0.961i)2-s + (−0.183 − 0.982i)3-s + (−0.850 + 0.526i)4-s + (0.961 + 0.273i)5-s + (0.895 − 0.445i)6-s + (−0.0922 + 0.995i)7-s + (−0.739 − 0.673i)8-s + (−0.932 + 0.361i)9-s + i·10-s + (0.850 − 0.526i)11-s + (0.673 + 0.739i)12-s + (0.995 + 0.0922i)13-s + (−0.982 + 0.183i)14-s + (0.0922 − 0.995i)15-s + (0.445 − 0.895i)16-s + (−0.739 + 0.673i)17-s + ⋯
L(s,χ)  = 1  + (0.273 + 0.961i)2-s + (−0.183 − 0.982i)3-s + (−0.850 + 0.526i)4-s + (0.961 + 0.273i)5-s + (0.895 − 0.445i)6-s + (−0.0922 + 0.995i)7-s + (−0.739 − 0.673i)8-s + (−0.932 + 0.361i)9-s + i·10-s + (0.850 − 0.526i)11-s + (0.673 + 0.739i)12-s + (0.995 + 0.0922i)13-s + (−0.982 + 0.183i)14-s + (0.0922 − 0.995i)15-s + (0.445 − 0.895i)16-s + (−0.739 + 0.673i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.449 + 0.893i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.449 + 0.893i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(137\)
\( \varepsilon \)  =  $0.449 + 0.893i$
motivic weight  =  \(0\)
character  :  $\chi_{137} (129, \cdot )$
Sato-Tate  :  $\mu(68)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 137,\ (0:\ ),\ 0.449 + 0.893i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.050680812 + 0.6477462349i$
$L(\frac12,\chi)$  $\approx$  $1.050680812 + 0.6477462349i$
$L(\chi,1)$  $\approx$  1.103739747 + 0.4387916953i
$L(1,\chi)$  $\approx$  1.103739747 + 0.4387916953i

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

−28.44263018908058431170243717632, −27.525916970840375621860967759700, −26.58407519091643531505333801791, −25.61513052452389059794677728290, −24.11356053329890651655341101396, −22.77233119802382082388764303045, −22.34624547948749300594368975046, −21.13810669959412463362978730732, −20.48437231718874535938998953655, −19.79331334198679906299054531402, −18.02865014502738554475827865748, −17.32396646713927300114871923664, −16.19549037882883455905592308082, −14.68038966359152195725681997536, −13.81812043523539439274670860624, −12.89400648381929627333931506194, −11.3247871676648911022415992123, −10.65091320894134863286255941539, −9.49880853800904237328538199711, −8.96880424512111358813894607839, −6.58455977696924836743295900420, −5.186097861251758635862057969191, −4.27973322416900175920550033274, −3.066788928966200977120360290518, −1.274506003190145530913782576210, 1.750052715616100354849405751686, 3.4316143080861803070154705994, 5.59237572218287518052635844796, 6.02501802716225839707570284402, 7.04773481527938297878386188269, 8.53529008074839481621394094868, 9.23534876103424628335094918308, 11.19268329556744385529000700350, 12.46420321944944421871919418223, 13.40483252356175891068315126039, 14.1611986012249640681912392123, 15.2631012880261769895732218002, 16.66073368389530520612183334817, 17.48791880951145502966801712854, 18.42744088086596202054006658575, 19.04168990642909985868290190222, 20.95058929265420837156180508806, 22.12941978839640144827844107260, 22.643182430431965120661602393975, 24.014976640959937834205912452168, 24.76458167425045780460382503033, 25.3980469644521838339834239332, 26.20366271932351818323619872531, 27.693541472070080299514721757487, 28.73114483311955976274052363320

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