Properties

Degree 1
Conductor 67
Sign $-0.817 + 0.575i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.235 + 0.971i)2-s + (−0.142 + 0.989i)3-s + (−0.888 + 0.458i)4-s + (0.415 + 0.909i)5-s + (−0.995 + 0.0950i)6-s + (0.723 − 0.690i)7-s + (−0.654 − 0.755i)8-s + (−0.959 − 0.281i)9-s + (−0.786 + 0.618i)10-s + (−0.995 − 0.0950i)11-s + (−0.327 − 0.945i)12-s + (0.981 + 0.189i)13-s + (0.841 + 0.540i)14-s + (−0.959 + 0.281i)15-s + (0.580 − 0.814i)16-s + (−0.888 − 0.458i)17-s + ⋯
L(s,χ)  = 1  + (0.235 + 0.971i)2-s + (−0.142 + 0.989i)3-s + (−0.888 + 0.458i)4-s + (0.415 + 0.909i)5-s + (−0.995 + 0.0950i)6-s + (0.723 − 0.690i)7-s + (−0.654 − 0.755i)8-s + (−0.959 − 0.281i)9-s + (−0.786 + 0.618i)10-s + (−0.995 − 0.0950i)11-s + (−0.327 − 0.945i)12-s + (0.981 + 0.189i)13-s + (0.841 + 0.540i)14-s + (−0.959 + 0.281i)15-s + (0.580 − 0.814i)16-s + (−0.888 − 0.458i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.817 + 0.575i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.817 + 0.575i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(67\)
\( \varepsilon \)  =  $-0.817 + 0.575i$
motivic weight  =  \(0\)
character  :  $\chi_{67} (36, \cdot )$
Sato-Tate  :  $\mu(33)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 67,\ (0:\ ),\ -0.817 + 0.575i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2884820658 + 0.9114103149i$
$L(\frac12,\chi)$  $\approx$  $0.2884820658 + 0.9114103149i$
$L(\chi,1)$  $\approx$  0.6678947325 + 0.8143118132i
$L(1,\chi)$  $\approx$  0.6678947325 + 0.8143118132i

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

−31.072215941131832954300571351164, −30.71657990919544866065795051666, −29.209348644488295727959056500533, −28.56195084111555761246956727956, −27.82868248586845265255385029432, −26.024787712920128160355323224044, −24.558091270048403525699035168807, −23.91122244537676710025160393232, −22.68985391804020791581836630699, −21.24453702679536420229912918552, −20.53109256967037393225416052933, −19.23263371503041840683695096722, −18.081785209628350144477922461443, −17.4714349435591119390270327138, −15.41956549682292558150207492699, −13.64667566754372953651267256381, −13.09691473771836549790751431608, −11.93933415327652478717495274983, −10.903616844544419208612815059637, −9.049683520227066552627329122783, −8.14359958639314541102562392393, −5.861955450989627440029148144196, −4.87783103538475420854515385381, −2.57499843491818646700586709229, −1.314183595044849075674615587277, 3.267556022665198318934960736926, 4.67462841248990696478683253655, 5.89657167522799137299079745200, 7.33589700801829443430595845375, 8.7693936583672747351684446486, 10.23719715390342219200858371866, 11.22680279961390365267788020569, 13.464772703612861363847207046925, 14.328218919346797924357821626049, 15.37819149828398708382980134907, 16.38495156988633129512174418780, 17.62215209396258664450718380651, 18.44588693242098661951561117419, 20.65724508305127415616973199117, 21.46531937283965626369433093573, 22.711225209415208150703825202705, 23.348301959624050906858786866715, 24.84102979122802530726117738592, 26.33854494686870609942533582252, 26.457479837347032564354167795068, 27.671707287029680646014109844982, 29.12737966485207980214082624378, 30.70187537609046279028986208248, 31.4415405587959891022513193496, 33.00291998581778273535357052273

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