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Properties

Label 2-670-67.64-c1-0-13
Degree $2$
Conductor $670$
Sign $0.997 - 0.0773i$
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 + (0.221 − 1.54i)3-s + (0.841 + 0.540i)4-s + (0.415 + 0.909i)5-s + (0.647 − 1.41i)6-s + (3.39 + 0.996i)7-s + (0.654 + 0.755i)8-s + (0.548 + 0.161i)9-s + (0.142 + 0.989i)10-s + (0.132 + 0.289i)11-s + (1.02 − 1.17i)12-s + (−2.45 + 2.83i)13-s + (2.97 + 1.91i)14-s + (1.49 − 0.438i)15-s + (0.415 + 0.909i)16-s + (−2.71 + 1.74i)17-s + ⋯
L(s)  = 1  + (0.678 + 0.199i)2-s + (0.128 − 0.890i)3-s + (0.420 + 0.270i)4-s + (0.185 + 0.406i)5-s + (0.264 − 0.578i)6-s + (1.28 + 0.376i)7-s + (0.231 + 0.267i)8-s + (0.182 + 0.0537i)9-s + (0.0450 + 0.313i)10-s + (0.0399 + 0.0874i)11-s + (0.294 − 0.339i)12-s + (−0.680 + 0.785i)13-s + (0.794 + 0.510i)14-s + (0.386 − 0.113i)15-s + (0.103 + 0.227i)16-s + (−0.657 + 0.422i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.997 - 0.0773i$
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: \((2,\ 670,\ (\ :1/2),\ 0.997 - 0.0773i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.71020 + 0.104954i\)
\(L(\frac12)\) \(\approx\) \(2.71020 + 0.104954i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-7.82 + 2.40i)T \)
good3 \( 1 + (-0.221 + 1.54i)T + (-2.87 - 0.845i)T^{2} \)
7 \( 1 + (-3.39 - 0.996i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (-0.132 - 0.289i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (2.45 - 2.83i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (2.71 - 1.74i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (1.13 - 0.331i)T + (15.9 - 10.2i)T^{2} \)
23 \( 1 + (-0.542 + 3.77i)T + (-22.0 - 6.47i)T^{2} \)
29 \( 1 - 5.05T + 29T^{2} \)
31 \( 1 + (5.82 + 6.71i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 - 6.85T + 37T^{2} \)
41 \( 1 + (4.03 - 2.59i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (-1.89 + 1.22i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 + (-0.283 + 1.97i)T + (-45.0 - 13.2i)T^{2} \)
53 \( 1 + (9.96 + 6.40i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (2.72 + 3.14i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (-0.997 + 2.18i)T + (-39.9 - 46.1i)T^{2} \)
71 \( 1 + (8.23 + 5.29i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (0.552 - 1.20i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (4.84 - 5.59i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (-5.72 - 12.5i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (2.47 + 17.1i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−10.84455325789048915006906242329, −9.655220428425701341799297630302, −8.426057155017220783608673003585, −7.77844742447806258079888288948, −6.88635207645769678478128508445, −6.22316713606229957433051957533, −4.95484332803901857083411701710, −4.21579925527124804513558455904, −2.44994779769621676946157871492, −1.78466741075259650430786012357, 1.45696219198519267352188136873, 2.95941296536023095519874533614, 4.24847937637634851570104069220, 4.79743604896411978076372433757, 5.53287677777221556454197123038, 6.96904143265350074573134435852, 7.87086963776844901999705497546, 8.914999065913214680170718705711, 9.788195112142838297632006089434, 10.64437833056907715616574694689

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