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Properties

Label 2-670-67.9-c1-0-16
Degree $2$
Conductor $670$
Sign $0.720 - 0.693i$
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  + (0.841 + 0.540i)2-s + (2.62 + 0.771i)3-s + (0.415 + 0.909i)4-s + (0.654 − 0.755i)5-s + (1.79 + 2.07i)6-s + (−1.38 − 0.889i)7-s + (−0.142 + 0.989i)8-s + (3.78 + 2.43i)9-s + (0.959 − 0.281i)10-s + (2.39 − 2.75i)11-s + (0.389 + 2.71i)12-s + (0.119 + 0.829i)13-s + (−0.683 − 1.49i)14-s + (2.30 − 1.48i)15-s + (−0.654 + 0.755i)16-s + (−0.383 + 0.840i)17-s + ⋯
L(s)  = 1  + (0.594 + 0.382i)2-s + (1.51 + 0.445i)3-s + (0.207 + 0.454i)4-s + (0.292 − 0.337i)5-s + (0.732 + 0.845i)6-s + (−0.522 − 0.336i)7-s + (−0.0503 + 0.349i)8-s + (1.26 + 0.811i)9-s + (0.303 − 0.0890i)10-s + (0.721 − 0.832i)11-s + (0.112 + 0.782i)12-s + (0.0330 + 0.229i)13-s + (−0.182 − 0.399i)14-s + (0.594 − 0.382i)15-s + (−0.163 + 0.188i)16-s + (−0.0930 + 0.203i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.720 - 0.693i$
Analytic conductor: \(5.34997\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{670} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ 0.720 - 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.13784 + 1.26536i\)
\(L(\frac12)\) \(\approx\) \(3.13784 + 1.26536i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.841 - 0.540i)T \)
5 \( 1 + (-0.654 + 0.755i)T \)
67 \( 1 + (7.23 + 3.82i)T \)
good3 \( 1 + (-2.62 - 0.771i)T + (2.52 + 1.62i)T^{2} \)
7 \( 1 + (1.38 + 0.889i)T + (2.90 + 6.36i)T^{2} \)
11 \( 1 + (-2.39 + 2.75i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (-0.119 - 0.829i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (0.383 - 0.840i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (3.13 - 2.01i)T + (7.89 - 17.2i)T^{2} \)
23 \( 1 + (-3.57 - 1.04i)T + (19.3 + 12.4i)T^{2} \)
29 \( 1 + 6.05T + 29T^{2} \)
31 \( 1 + (0.768 - 5.34i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + 4.79T + 37T^{2} \)
41 \( 1 + (-0.992 + 2.17i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (0.0134 - 0.0294i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 + (4.71 + 1.38i)T + (39.5 + 25.4i)T^{2} \)
53 \( 1 + (1.88 + 4.11i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (-1.07 + 7.48i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (4.23 + 4.88i)T + (-8.68 + 60.3i)T^{2} \)
71 \( 1 + (-1.33 - 2.91i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-9.54 - 11.0i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (0.115 + 0.800i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (5.04 - 5.81i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-10.0 + 2.95i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−10.44024360802782197230675795021, −9.465711631931637298358510243911, −8.827154448852795366709115250903, −8.193756617811548059213006409342, −7.09321876811848297358002294778, −6.20426249849903126538740451245, −4.96148398737434979821554190838, −3.76247551844092484387444182664, −3.35980267118696003006282086521, −1.91726705114640842367254946957, 1.76490617105689879353317113614, 2.64623383925256793055445582043, 3.50216518281398440306117909578, 4.57179770035956839370486978453, 6.04165940131389385682324601775, 6.93012504530117500495500167399, 7.67690103299520551921335630515, 8.999252750132742103592605551650, 9.317081485112497557843656760897, 10.28214506037420952118958932917

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