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Properties

Label 2-670-67.9-c1-0-0
Degree $2$
Conductor $670$
Sign $-0.970 - 0.239i$
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 + (−1.68 − 0.494i)3-s + (0.415 + 0.909i)4-s + (0.654 − 0.755i)5-s + (−1.14 − 1.32i)6-s + (−2.93 − 1.88i)7-s + (−0.142 + 0.989i)8-s + (0.0674 + 0.0433i)9-s + (0.959 − 0.281i)10-s + (0.584 − 0.674i)11-s + (−0.249 − 1.73i)12-s + (0.786 + 5.46i)13-s + (−1.44 − 3.17i)14-s + (−1.47 + 0.948i)15-s + (−0.654 + 0.755i)16-s + (−2.82 + 6.19i)17-s + ⋯
L(s)  = 1  + (0.594 + 0.382i)2-s + (−0.972 − 0.285i)3-s + (0.207 + 0.454i)4-s + (0.292 − 0.337i)5-s + (−0.469 − 0.541i)6-s + (−1.10 − 0.713i)7-s + (−0.0503 + 0.349i)8-s + (0.0224 + 0.0144i)9-s + (0.303 − 0.0890i)10-s + (0.176 − 0.203i)11-s + (−0.0721 − 0.501i)12-s + (0.218 + 1.51i)13-s + (−0.387 − 0.848i)14-s + (−0.381 + 0.244i)15-s + (−0.163 + 0.188i)16-s + (−0.685 + 1.50i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $-0.970 - 0.239i$
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: \((2,\ 670,\ (\ :1/2),\ -0.970 - 0.239i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0338229 + 0.278416i\)
\(L(\frac12)\) \(\approx\) \(0.0338229 + 0.278416i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.626 - 8.16i)T \)
good3 \( 1 + (1.68 + 0.494i)T + (2.52 + 1.62i)T^{2} \)
7 \( 1 + (2.93 + 1.88i)T + (2.90 + 6.36i)T^{2} \)
11 \( 1 + (-0.584 + 0.674i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (-0.786 - 5.46i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (2.82 - 6.19i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (5.06 - 3.25i)T + (7.89 - 17.2i)T^{2} \)
23 \( 1 + (6.06 + 1.78i)T + (19.3 + 12.4i)T^{2} \)
29 \( 1 + 0.877T + 29T^{2} \)
31 \( 1 + (-1.37 + 9.56i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + 6.89T + 37T^{2} \)
41 \( 1 + (0.545 - 1.19i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (0.231 - 0.506i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 + (-3.48 - 1.02i)T + (39.5 + 25.4i)T^{2} \)
53 \( 1 + (-1.39 - 3.06i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (0.669 - 4.65i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (8.74 + 10.0i)T + (-8.68 + 60.3i)T^{2} \)
71 \( 1 + (1.94 + 4.25i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-0.881 - 1.01i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (-0.851 - 5.92i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (-4.31 + 4.98i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (1.67 - 0.492i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 - 1.49T + 97T^{2} \)
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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

−10.98630297305784275643357774408, −10.23742909147780478974692041061, −9.141802053568134027147112576840, −8.230184958603390525548623792034, −6.89576371207635649423069527524, −6.19151578357809916376206547745, −6.00423159686956577521545170762, −4.35489461280743576348434896203, −3.82101808563634517881439530928, −1.91682572713813667950062825421, 0.12755019860819651279904347010, 2.44197250370901680155569914332, 3.30732395000020324748495026541, 4.76886068148205728811282190341, 5.55013460782111154044510556273, 6.25493254991322449029432868694, 7.01891215148638174126554234552, 8.581444224005028883137070769195, 9.543888630053989513320835190306, 10.42557326951574320386975307428

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