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Properties

Label 2-670-67.14-c1-0-10
Degree $2$
Conductor $670$
Sign $0.945 - 0.324i$
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  + (−0.415 − 0.909i)2-s + (2.36 + 1.52i)3-s + (−0.654 + 0.755i)4-s + (−0.142 − 0.989i)5-s + (0.400 − 2.78i)6-s + (1.07 + 2.34i)7-s + (0.959 + 0.281i)8-s + (2.03 + 4.46i)9-s + (−0.841 + 0.540i)10-s + (−0.314 − 2.18i)11-s + (−2.69 + 0.792i)12-s + (1.27 − 0.374i)13-s + (1.68 − 1.94i)14-s + (1.16 − 2.55i)15-s + (−0.142 − 0.989i)16-s + (1.42 + 1.64i)17-s + ⋯
L(s)  = 1  + (−0.293 − 0.643i)2-s + (1.36 + 0.877i)3-s + (−0.327 + 0.377i)4-s + (−0.0636 − 0.442i)5-s + (0.163 − 1.13i)6-s + (0.404 + 0.886i)7-s + (0.339 + 0.0996i)8-s + (0.679 + 1.48i)9-s + (−0.266 + 0.170i)10-s + (−0.0947 − 0.658i)11-s + (−0.779 + 0.228i)12-s + (0.353 − 0.103i)13-s + (0.451 − 0.520i)14-s + (0.301 − 0.660i)15-s + (−0.0355 − 0.247i)16-s + (0.345 + 0.398i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 - 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.945 - 0.324i$
Analytic conductor: \(5.34997\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{670} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ 0.945 - 0.324i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02681 + 0.338313i\)
\(L(\frac12)\) \(\approx\) \(2.02681 + 0.338313i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 + 0.909i)T \)
5 \( 1 + (0.142 + 0.989i)T \)
67 \( 1 + (3.46 - 7.41i)T \)
good3 \( 1 + (-2.36 - 1.52i)T + (1.24 + 2.72i)T^{2} \)
7 \( 1 + (-1.07 - 2.34i)T + (-4.58 + 5.29i)T^{2} \)
11 \( 1 + (0.314 + 2.18i)T + (-10.5 + 3.09i)T^{2} \)
13 \( 1 + (-1.27 + 0.374i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (-1.42 - 1.64i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (0.799 - 1.75i)T + (-12.4 - 14.3i)T^{2} \)
23 \( 1 + (-6.08 - 3.90i)T + (9.55 + 20.9i)T^{2} \)
29 \( 1 - 3.47T + 29T^{2} \)
31 \( 1 + (7.80 + 2.29i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + 1.25T + 37T^{2} \)
41 \( 1 + (-1.39 - 1.61i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (1.59 + 1.84i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + (-3.39 - 2.17i)T + (19.5 + 42.7i)T^{2} \)
53 \( 1 + (4.04 - 4.67i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (7.87 + 2.31i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (-0.644 + 4.48i)T + (-58.5 - 17.1i)T^{2} \)
71 \( 1 + (-3.29 + 3.79i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (-2.23 + 15.5i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (10.5 - 3.09i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (0.761 + 5.29i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (-11.9 + 7.69i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 - 9.16T + 97T^{2} \)
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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

−10.47176302615681776332197006683, −9.415970729205095229176292818159, −8.957435755633518797704788762147, −8.353734582230753546925107891666, −7.61337655190002605689647129053, −5.77253481152016582278850446328, −4.77171050619377729350210832216, −3.66852957702196952148962231028, −2.93616621355044737886365704932, −1.70119033048825027822672763648, 1.23406324553341397049014396947, 2.56709037824240597173153753425, 3.76261839363347168341423769468, 4.94891493079557979565799992775, 6.55445226195350716500670632108, 7.19831698817368876347966974181, 7.66005273760709332150481917535, 8.602158715761291104342655336167, 9.248523733030876203008766515675, 10.30640354068891543305916175275

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