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Properties

Label 2-670-67.64-c1-0-14
Degree $2$
Conductor $670$
Sign $0.794 + 0.607i$
Analytic cond. $5.34997$
Root an. cond. $2.31300$
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  + (0.959 + 0.281i)2-s + (0.251 − 1.74i)3-s + (0.841 + 0.540i)4-s + (0.415 + 0.909i)5-s + (0.732 − 1.60i)6-s + (−0.408 − 0.120i)7-s + (0.654 + 0.755i)8-s + (−0.108 − 0.0317i)9-s + (0.142 + 0.989i)10-s + (−1.34 − 2.95i)11-s + (1.15 − 1.33i)12-s + (0.852 − 0.983i)13-s + (−0.358 − 0.230i)14-s + (1.69 − 0.497i)15-s + (0.415 + 0.909i)16-s + (5.01 − 3.22i)17-s + ⋯
L(s)  = 1  + (0.678 + 0.199i)2-s + (0.144 − 1.00i)3-s + (0.420 + 0.270i)4-s + (0.185 + 0.406i)5-s + (0.299 − 0.655i)6-s + (−0.154 − 0.0453i)7-s + (0.231 + 0.267i)8-s + (−0.0360 − 0.0105i)9-s + (0.0450 + 0.313i)10-s + (−0.406 − 0.890i)11-s + (0.333 − 0.384i)12-s + (0.236 − 0.272i)13-s + (−0.0958 − 0.0615i)14-s + (0.437 − 0.128i)15-s + (0.103 + 0.227i)16-s + (1.21 − 0.781i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.794 + 0.607i$
Analytic conductor: \(5.34997\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{670} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ 0.794 + 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.36803 - 0.801071i\)
\(L(\frac12)\) \(\approx\) \(2.36803 - 0.801071i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.959 - 0.281i)T \)
5 \( 1 + (-0.415 - 0.909i)T \)
67 \( 1 + (-7.43 - 3.42i)T \)
good3 \( 1 + (-0.251 + 1.74i)T + (-2.87 - 0.845i)T^{2} \)
7 \( 1 + (0.408 + 0.120i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (1.34 + 2.95i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (-0.852 + 0.983i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (-5.01 + 3.22i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (-6.42 + 1.88i)T + (15.9 - 10.2i)T^{2} \)
23 \( 1 + (0.740 - 5.15i)T + (-22.0 - 6.47i)T^{2} \)
29 \( 1 + 6.66T + 29T^{2} \)
31 \( 1 + (-1.91 - 2.20i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + 7.38T + 37T^{2} \)
41 \( 1 + (5.28 - 3.39i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (-2.52 + 1.62i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 + (-0.330 + 2.29i)T + (-45.0 - 13.2i)T^{2} \)
53 \( 1 + (-9.48 - 6.09i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (-3.93 - 4.54i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (4.13 - 9.05i)T + (-39.9 - 46.1i)T^{2} \)
71 \( 1 + (12.0 + 7.75i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (2.96 - 6.48i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (3.75 - 4.33i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (4.13 + 9.04i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-2.26 - 15.7i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 - 0.475T + 97T^{2} \)
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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

−10.51625123919094536763085395832, −9.621749140766967444936719745584, −8.370806085523956911982928478052, −7.39730563929396921517084457588, −7.11104714414467316764708106137, −5.81386265255458194087143150032, −5.28937857947242504974060151788, −3.55607251072676264983665491971, −2.80562458396070526545410702197, −1.27903348983039635953863476326, 1.68763416843878127056452667098, 3.26110557704820184873094568457, 4.06675246914695875789439949210, 5.02169825291225263324929321017, 5.71627224160454626371791675456, 6.98973289852551177888987129821, 7.997068541040951017908543950753, 9.171233693417639556147661502050, 9.995001608833374744728424037528, 10.31447624599767279296043616591

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