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Properties

Label 2-67-67.62-c1-0-2
Degree $2$
Conductor $67$
Sign $0.855 + 0.518i$
Analytic cond. $0.534997$
Root an. cond. $0.731435$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.10 + 1.27i)2-s + (−1.28 − 2.81i)3-s + (−0.118 − 0.822i)4-s + (2.46 − 0.722i)5-s + (5.00 + 1.46i)6-s + (2.41 − 2.78i)7-s + (−1.65 − 1.06i)8-s + (−4.32 + 4.99i)9-s + (−1.79 + 3.92i)10-s + (−1.90 + 0.558i)11-s + (−2.16 + 1.39i)12-s + (1.22 − 0.785i)13-s + (0.883 + 6.14i)14-s + (−5.20 − 6.00i)15-s + (4.77 − 1.40i)16-s + (−0.713 + 4.96i)17-s + ⋯
L(s)  = 1  + (−0.779 + 0.899i)2-s + (−0.743 − 1.62i)3-s + (−0.0591 − 0.411i)4-s + (1.10 − 0.323i)5-s + (2.04 + 0.599i)6-s + (0.913 − 1.05i)7-s + (−0.585 − 0.376i)8-s + (−1.44 + 1.66i)9-s + (−0.566 + 1.24i)10-s + (−0.573 + 0.168i)11-s + (−0.625 + 0.401i)12-s + (0.338 − 0.217i)13-s + (0.236 + 1.64i)14-s + (−1.34 − 1.55i)15-s + (1.19 − 0.350i)16-s + (−0.173 + 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(67\)
Sign: $0.855 + 0.518i$
Analytic conductor: \(0.534997\)
Root analytic conductor: \(0.731435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 67,\ (\ :1/2),\ 0.855 + 0.518i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.578586 - 0.161627i\)
\(L(\frac12)\) \(\approx\) \(0.578586 - 0.161627i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 + (4.70 - 6.69i)T \)
good2 \( 1 + (1.10 - 1.27i)T + (-0.284 - 1.97i)T^{2} \)
3 \( 1 + (1.28 + 2.81i)T + (-1.96 + 2.26i)T^{2} \)
5 \( 1 + (-2.46 + 0.722i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (-2.41 + 2.78i)T + (-0.996 - 6.92i)T^{2} \)
11 \( 1 + (1.90 - 0.558i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (-1.22 + 0.785i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (0.713 - 4.96i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-1.45 - 1.68i)T + (-2.70 + 18.8i)T^{2} \)
23 \( 1 + (-1.18 - 2.58i)T + (-15.0 + 17.3i)T^{2} \)
29 \( 1 - 1.58T + 29T^{2} \)
31 \( 1 + (1.45 + 0.932i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 - 2.65T + 37T^{2} \)
41 \( 1 + (1.18 - 8.25i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (0.415 - 2.88i)T + (-41.2 - 12.1i)T^{2} \)
47 \( 1 + (1.42 + 3.12i)T + (-30.7 + 35.5i)T^{2} \)
53 \( 1 + (1.94 + 13.5i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (1.80 + 1.15i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (-9.60 - 2.82i)T + (51.3 + 32.9i)T^{2} \)
71 \( 1 + (0.520 + 3.61i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (5.81 + 1.70i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (11.7 - 7.53i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (-11.6 + 3.41i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (-2.99 + 6.55i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + 15.5T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.63920638017689312945895355236, −13.40688579655886602141541723350, −12.86476758729160933488820774374, −11.41331916516023577755723962949, −10.11718916121335849432436827370, −8.324538744622088608830318840059, −7.61116594512485716201343017966, −6.50681127382633504702372150639, −5.50521564389646233081484978556, −1.44204479357260207131182942422, 2.63819765820075461372038159812, 5.04562977079289894648973466351, 5.87774747377419348942514375319, 8.807192024039260679705635849121, 9.468242099082880200759079228890, 10.45817322154131473034826715851, 11.15828511975703503249578084484, 12.00155206931976358847645645171, 14.07801906681318165651171929604, 15.10313218964818636366732356315

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