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Properties

Label 2-670-67.29-c1-0-11
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: $\chi_{670} (431, \cdot )$
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

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

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