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Properties

Label 1-67-67.36-r0-0-0
Degree $1$
Conductor $67$
Sign $-0.817 + 0.575i$
Analytic cond. $0.311146$
Root an. cond. $0.311146$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $-0.817 + 0.575i$
Analytic conductor: \(0.311146\)
Root analytic conductor: \(0.311146\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (0:\ ),\ -0.817 + 0.575i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2884820658 + 0.9114103149i\)
\(L(\frac12)\) \(\approx\) \(0.2884820658 + 0.9114103149i\)
\(L(1)\) \(\approx\) \(0.6678947325 + 0.8143118132i\)
\(L(1)\) \(\approx\) \(0.6678947325 + 0.8143118132i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (0.235 + 0.971i)T \)
3 \( 1 + (-0.142 + 0.989i)T \)
5 \( 1 + (0.415 + 0.909i)T \)
7 \( 1 + (0.723 - 0.690i)T \)
11 \( 1 + (-0.995 - 0.0950i)T \)
13 \( 1 + (0.981 + 0.189i)T \)
17 \( 1 + (-0.888 - 0.458i)T \)
19 \( 1 + (0.723 + 0.690i)T \)
23 \( 1 + (0.928 - 0.371i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.981 - 0.189i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.0475 + 0.998i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (-0.786 - 0.618i)T \)
53 \( 1 + (0.841 + 0.540i)T \)
59 \( 1 + (-0.654 - 0.755i)T \)
61 \( 1 + (-0.995 + 0.0950i)T \)
71 \( 1 + (-0.888 + 0.458i)T \)
73 \( 1 + (-0.995 + 0.0950i)T \)
79 \( 1 + (-0.327 - 0.945i)T \)
83 \( 1 + (0.580 - 0.814i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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