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Properties

Label 2-670-67.24-c1-0-5
Degree $2$
Conductor $670$
Sign $0.516 - 0.856i$
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 + (−0.351 + 0.225i)3-s + (−0.654 − 0.755i)4-s + (0.142 − 0.989i)5-s + (0.0594 + 0.413i)6-s + (−0.689 + 1.50i)7-s + (−0.959 + 0.281i)8-s + (−1.17 + 2.57i)9-s + (−0.841 − 0.540i)10-s + (−0.645 + 4.49i)11-s + (0.400 + 0.117i)12-s + (0.744 + 0.218i)13-s + (1.08 + 1.25i)14-s + (0.173 + 0.380i)15-s + (−0.142 + 0.989i)16-s + (−5.26 + 6.07i)17-s + ⋯
L(s)  = 1  + (0.293 − 0.643i)2-s + (−0.202 + 0.130i)3-s + (−0.327 − 0.377i)4-s + (0.0636 − 0.442i)5-s + (0.0242 + 0.168i)6-s + (−0.260 + 0.570i)7-s + (−0.339 + 0.0996i)8-s + (−0.391 + 0.856i)9-s + (−0.266 − 0.170i)10-s + (−0.194 + 1.35i)11-s + (0.115 + 0.0339i)12-s + (0.206 + 0.0606i)13-s + (0.290 + 0.335i)14-s + (0.0448 + 0.0981i)15-s + (−0.0355 + 0.247i)16-s + (−1.27 + 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.516 - 0.856i$
Analytic conductor: \(5.34997\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{670} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ 0.516 - 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.923735 + 0.521847i\)
\(L(\frac12)\) \(\approx\) \(0.923735 + 0.521847i\)
\(L(\frac{3}{2})\) 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 \( 1 + (7.97 - 1.82i)T \)
good3 \( 1 + (0.351 - 0.225i)T + (1.24 - 2.72i)T^{2} \)
7 \( 1 + (0.689 - 1.50i)T + (-4.58 - 5.29i)T^{2} \)
11 \( 1 + (0.645 - 4.49i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (-0.744 - 0.218i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (5.26 - 6.07i)T + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (0.555 + 1.21i)T + (-12.4 + 14.3i)T^{2} \)
23 \( 1 + (-5.95 + 3.82i)T + (9.55 - 20.9i)T^{2} \)
29 \( 1 - 8.17T + 29T^{2} \)
31 \( 1 + (6.70 - 1.96i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 - 2.03T + 37T^{2} \)
41 \( 1 + (7.46 - 8.61i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (1.57 - 1.81i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + (-5.92 + 3.80i)T + (19.5 - 42.7i)T^{2} \)
53 \( 1 + (1.47 + 1.70i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (0.562 - 0.165i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (-0.293 - 2.04i)T + (-58.5 + 17.1i)T^{2} \)
71 \( 1 + (1.99 + 2.30i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-2.30 - 16.0i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (9.58 + 2.81i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (-0.988 + 6.87i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (-4.02 - 2.58i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 - 2.79T + 97T^{2} \)
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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

−10.66956648534414743835297379795, −10.02496567738218406913168627909, −8.887171143036141615380357556376, −8.417495511227842848463030455004, −6.98828540050509864600916388490, −5.99219113306447261502111131772, −4.89251072918280156679503522046, −4.38356339563891572140308408656, −2.76366488434709903613401075042, −1.79586918175190622001534172088, 0.51803606842156463566212523908, 2.92044870563249907270267007800, 3.71352382736542682607542001580, 5.06390910780535653817986875656, 6.01998895526629761375606813438, 6.77204074301744615310550674683, 7.43631700070451789646374586952, 8.742756232227061035456212173250, 9.191771309653900880076674365367, 10.50374830095429226722456611243

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