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Properties

Label 2-670-335.104-c1-0-30
Degree $2$
Conductor $670$
Sign $-0.523 + 0.851i$
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  + (0.866 − 0.5i)2-s − 0.0712i·3-s + (0.499 − 0.866i)4-s + (0.722 − 2.11i)5-s + (−0.0356 − 0.0617i)6-s + (−3.87 − 2.23i)7-s − 0.999i·8-s + 2.99·9-s + (−0.431 − 2.19i)10-s + (−1.43 + 2.49i)11-s + (−0.0617 − 0.0356i)12-s + (4.80 − 2.77i)13-s − 4.47·14-s + (−0.150 − 0.0515i)15-s + (−0.5 − 0.866i)16-s + (−4.09 + 2.36i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s − 0.0411i·3-s + (0.249 − 0.433i)4-s + (0.323 − 0.946i)5-s + (−0.0145 − 0.0252i)6-s + (−1.46 − 0.846i)7-s − 0.353i·8-s + 0.998·9-s + (−0.136 − 0.693i)10-s + (−0.434 + 0.751i)11-s + (−0.0178 − 0.0102i)12-s + (1.33 − 0.768i)13-s − 1.19·14-s + (−0.0389 − 0.0133i)15-s + (−0.125 − 0.216i)16-s + (−0.993 + 0.573i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.523 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $-0.523 + 0.851i$
Analytic conductor: \(5.34997\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{670} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ -0.523 + 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.903961 - 1.61673i\)
\(L(\frac12)\) \(\approx\) \(0.903961 - 1.61673i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-0.722 + 2.11i)T \)
67 \( 1 + (-7.75 - 2.61i)T \)
good3 \( 1 + 0.0712iT - 3T^{2} \)
7 \( 1 + (3.87 + 2.23i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.43 - 2.49i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.80 + 2.77i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.09 - 2.36i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.42 + 4.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.63 - 2.10i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.23 + 5.60i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.67 - 2.90i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.30 + 3.64i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.932 + 1.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 8.54iT - 43T^{2} \)
47 \( 1 + (-8.62 - 4.98i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.70iT - 53T^{2} \)
59 \( 1 - 9.20T + 59T^{2} \)
61 \( 1 + (-4.03 - 6.99i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (2.78 - 4.81i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.20 - 2.42i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.69 + 8.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.69 - 2.13i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.04T + 89T^{2} \)
97 \( 1 + (13.9 - 8.02i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.25382749909266940019314353745, −9.609205664758976979617862328859, −8.641982032271649603892411925279, −7.37857910797118457114010169383, −6.49992244683694540522485465231, −5.70573840117002962270023312077, −4.31287532742100191388745188795, −3.91868705690155271382138281488, −2.32470479653114722533055257979, −0.812622635080025363719220685725, 2.24102743223872467107698844639, 3.29720497750318502870494588560, 4.14421548930607079990366748902, 5.70710273628990479159969747907, 6.45117709668795330580793048686, 6.74466771056152103046071826934, 8.138637762601199431093228136106, 9.120829008696758919567026772996, 9.963906939839892511947005129128, 10.82034374630649679520241709017

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