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Properties

Label 2-670-67.37-c1-0-10
Degree $2$
Conductor $670$
Sign $0.992 - 0.124i$
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 + 3.44·3-s + (−0.499 + 0.866i)4-s − 5-s + (−1.72 − 2.98i)6-s + (−2.22 + 3.85i)7-s + 0.999·8-s + 8.89·9-s + (0.5 + 0.866i)10-s + (−1.72 + 2.98i)12-s + (2 + 3.46i)13-s + 4.44·14-s − 3.44·15-s + (−0.5 − 0.866i)16-s + (−2.22 − 3.85i)17-s + (−4.44 − 7.70i)18-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + 1.99·3-s + (−0.249 + 0.433i)4-s − 0.447·5-s + (−0.704 − 1.21i)6-s + (−0.840 + 1.45i)7-s + 0.353·8-s + 2.96·9-s + (0.158 + 0.273i)10-s + (−0.497 + 0.862i)12-s + (0.554 + 0.960i)13-s + 1.18·14-s − 0.890·15-s + (−0.125 − 0.216i)16-s + (−0.539 − 0.934i)17-s + (−1.04 − 1.81i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.992 - 0.124i$
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: \((2,\ 670,\ (\ :1/2),\ 0.992 - 0.124i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14268 + 0.133822i\)
\(L(\frac12)\) \(\approx\) \(2.14268 + 0.133822i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.174 + 8.18i)T \)
good3 \( 1 - 3.44T + 3T^{2} \)
7 \( 1 + (2.22 - 3.85i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.22 + 3.85i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.44 - 5.97i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.22 + 2.12i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.275 + 0.476i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.94 + 5.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.05 + 1.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (3.67 - 6.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7T + 53T^{2} \)
59 \( 1 + 6.89T + 59T^{2} \)
61 \( 1 + (2.22 + 3.85i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (0.174 - 0.301i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.22 - 7.31i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.17 + 10.6i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.10T + 89T^{2} \)
97 \( 1 + (7.89 + 13.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.07670978860277359945282190588, −9.399436664993338590513184205589, −8.968227702289004046111535987566, −8.238559652144531192466389877219, −7.43157624036432485288988010844, −6.28854812154209209303243185526, −4.48872588251248849019685069830, −3.50970454211007370878836887646, −2.75758467539322713591692858938, −1.85582156206741149761468193412, 1.17665385958427678228729516431, 3.03480494911122620522159622307, 3.69842647550057319261697672299, 4.66546851966424850505807574852, 6.60422777428930369592280271486, 7.21562471405022640139786437512, 7.923005254564899065314851037889, 8.621442337812122333590187902351, 9.429823781066577187377768562837, 10.20561171529580682298310138598

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