↑

Properties

Label 2-670-67.29-c1-0-17
Degree $2$
Conductor $670$
Sign $-0.978 + 0.205i$
Analytic cond. $5.34997$
Root an. cond. $2.31300$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s − 3·3-s + (−0.499 − 0.866i)4-s + 5-s + (1.5 − 2.59i)6-s + (−2 − 3.46i)7-s + 0.999·8-s + 6·9-s + (−0.5 + 0.866i)10-s + (1 + 1.73i)11-s + (1.49 + 2.59i)12-s + (1 − 1.73i)13-s + 3.99·14-s − 3·15-s + (−0.5 + 0.866i)16-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s − 1.73·3-s + (−0.249 − 0.433i)4-s + 0.447·5-s + (0.612 − 1.06i)6-s + (−0.755 − 1.30i)7-s + 0.353·8-s + 2·9-s + (−0.158 + 0.273i)10-s + (0.301 + 0.522i)11-s + (0.433 + 0.749i)12-s + (0.277 − 0.480i)13-s + 1.06·14-s − 0.774·15-s + (−0.125 + 0.216i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $-0.978 + 0.205i$
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: \(1\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ -0.978 + 0.205i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 7.79i)T \)
good3 \( 1 + 3T + 3T^{2} \)
7 \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.5 - 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 + 10T + 59T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.5 - 4.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.15498606042256538505874726912, −9.680641878358550112200524944004, −8.131766904375894024292347320415, −7.12538808038979828048985024019, −6.40801607788018761661018700801, −5.93630572416031497976554396723, −4.78108654962873450571995251204, −3.89254160338476084249332484903, −1.37283499345235596117113844486, 0, 1.71279955416930133062366861460, 3.19385535078606659556448492216, 4.71868783496237793776723292871, 5.56365921478389656865839763787, 6.35188854956754173514078900195, 7.03902471684327675830145713401, 8.826418553295345423785705667941, 9.223610140804533042414699836210, 10.24053144123874295578454055282

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