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Properties

Label 2-67-67.40-c1-0-0
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: $\chi_{67} (40, \cdot )$
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

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

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