Properties

Label 2-670-67.37-c1-0-3
Degree $2$
Conductor $670$
Sign $-0.405 - 0.914i$
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  + (0.5 + 0.866i)2-s − 2.58·3-s + (−0.499 + 0.866i)4-s − 5-s + (−1.29 − 2.23i)6-s + (1.94 − 3.37i)7-s − 0.999·8-s + 3.67·9-s + (−0.5 − 0.866i)10-s + (−0.838 + 1.45i)11-s + (1.29 − 2.23i)12-s + (2.29 + 3.97i)13-s + 3.89·14-s + 2.58·15-s + (−0.5 − 0.866i)16-s + (−0.0811 − 0.140i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s − 1.49·3-s + (−0.249 + 0.433i)4-s − 0.447·5-s + (−0.527 − 0.913i)6-s + (0.735 − 1.27i)7-s − 0.353·8-s + 1.22·9-s + (−0.158 − 0.273i)10-s + (−0.252 + 0.437i)11-s + (0.372 − 0.646i)12-s + (0.636 + 1.10i)13-s + 1.04·14-s + 0.667·15-s + (−0.125 − 0.216i)16-s + (−0.0196 − 0.0340i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 - 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.405 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $-0.405 - 0.914i$
Analytic conductor: \(5.34997\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{670} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ -0.405 - 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.440202 + 0.677003i\)
\(L(\frac12)\) \(\approx\) \(0.440202 + 0.677003i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + T \)
67 \( 1 + (-7.78 + 2.53i)T \)
good3 \( 1 + 2.58T + 3T^{2} \)
7 \( 1 + (-1.94 + 3.37i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.838 - 1.45i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.29 - 3.97i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.0811 + 0.140i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.127 - 0.221i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.309 + 0.535i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.77 - 6.54i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.19 - 7.27i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.94 - 5.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.10 - 3.64i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 5.30T + 43T^{2} \)
47 \( 1 + (1.02 - 1.76i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.30T + 53T^{2} \)
59 \( 1 - 0.910T + 59T^{2} \)
61 \( 1 + (-5.89 - 10.2i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (0.520 - 0.900i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.08 - 3.61i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.92 + 8.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.62 - 6.27i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 2.79T + 89T^{2} \)
97 \( 1 + (-4.80 - 8.32i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.00672240515975026050374588756, −10.24826574265518453857587452347, −8.946297466589613097394335686343, −7.79976662916239374057520586428, −7.02773468872612300565376946342, −6.45509165234292902824870009383, −5.22046500220202603972948406613, −4.60508680258514269661130775540, −3.75434315799866711461025440614, −1.28447125059914305867765303885, 0.52953792955606889929686904938, 2.23128864360318074021740375013, 3.71737697528051002761084168573, 4.93259977668798860651569823462, 5.67764236259226906969797421075, 6.07349968083685616334309562895, 7.65088018118223313254092478239, 8.520736791966623160544313082680, 9.616512237629788548373138479512, 10.71803461664903451570599095384

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