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Properties

Label 1-67-67.37-r0-0-0
Degree $1$
Conductor $67$
Sign $0.809 + 0.586i$
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  + (−0.5 + 0.866i)2-s + 3-s + (−0.5 − 0.866i)4-s + 5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + 9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + 14-s + 15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + 3-s + (−0.5 − 0.866i)4-s + 5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + 9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + 14-s + 15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $0.809 + 0.586i$
Analytic conductor: \(0.311146\)
Root analytic conductor: \(0.311146\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (0:\ ),\ 0.809 + 0.586i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9566209793 + 0.3101595422i\)
\(L(\frac12)\) \(\approx\) \(0.9566209793 + 0.3101595422i\)
\(L(1)\) \(\approx\) \(1.045565782 + 0.2882887488i\)
\(L(1)\) \(\approx\) \(1.045565782 + 0.2882887488i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + T \)
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

−31.734328194530924450901411054, −30.71906597583668520902114750448, −29.69508230087233881826897467868, −28.7215592423847176556411639882, −27.65046795112191552896075524618, −26.25905399685517679059540037509, −25.57856611000383641819897831735, −24.72294888033272516354112372939, −22.48260534187109209618747055822, −21.63606946682300554271715377492, −20.5709863403771278849367228218, −19.74172351194684631510784897961, −18.39749727137024929085532699512, −17.7595508827905523201094815594, −15.97258899351231866932065098280, −14.55483288584328563021782049228, −13.113013033426080362887193859116, −12.54687392645402047709647639143, −10.48169567714137167019090875493, −9.52337119791475147862054394525, −8.709517189125195522418878190114, −7.12662258170354519885955712626, −4.90912131021594238312018548924, −2.87588070876532184435116229667, −2.14447304380519374943053029498, 1.879391647351747033046785612033, 4.02380268123247089792132075807, 5.923514614460286331214908418314, 7.16585894040888362850714231635, 8.46344299324580560564784425007, 9.62723045560199383447576005853, 10.45053281042900295348817677002, 13.25940344534530425188505490277, 13.81004673190599756721944420028, 14.9253429910191898558050804492, 16.33477733695046932587264806747, 17.2426800381579665198263859168, 18.699569791995021132613836596150, 19.48978408348497224722234890397, 20.87173863246160879150820704089, 22.09511016983960599023519882411, 23.80053845540282900836291043847, 24.5377999302475958567705160043, 25.9664000847639651593334957409, 26.11101556776264715867975610902, 27.29337104687190317621229553444, 28.94876328963653684096390016560, 29.74325188245316012894116868225, 31.44528209215799632571623817664, 32.339912999649138495952507859900

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