↑

Properties

Label 2-670-67.37-c1-0-18
Degree $2$
Conductor $670$
Sign $0.815 - 0.579i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + 2.08·3-s + (−0.499 + 0.866i)4-s + 5-s + (1.04 + 1.80i)6-s + (1.21 − 2.11i)7-s − 0.999·8-s + 1.35·9-s + (0.5 + 0.866i)10-s + (1.67 − 2.90i)11-s + (−1.04 + 1.80i)12-s + (1.21 + 2.11i)13-s + 2.43·14-s + 2.08·15-s + (−0.5 − 0.866i)16-s + (−0.367 − 0.635i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + 1.20·3-s + (−0.249 + 0.433i)4-s + 0.447·5-s + (0.425 + 0.737i)6-s + (0.460 − 0.798i)7-s − 0.353·8-s + 0.450·9-s + (0.158 + 0.273i)10-s + (0.505 − 0.875i)11-s + (−0.301 + 0.521i)12-s + (0.338 + 0.585i)13-s + 0.651·14-s + 0.538·15-s + (−0.125 − 0.216i)16-s + (−0.0890 − 0.154i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.815 - 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.815 - 0.579i$
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: \((2,\ 670,\ (\ :1/2),\ 0.815 - 0.579i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.72159 + 0.868722i\)
\(L(\frac12)\) \(\approx\) \(2.72159 + 0.868722i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.02 + 7.12i)T \)
good3 \( 1 - 2.08T + 3T^{2} \)
7 \( 1 + (-1.21 + 2.11i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.67 + 2.90i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.21 - 2.11i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.367 + 0.635i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.21 - 3.84i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.895 + 1.55i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.08 - 8.80i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.82 + 10.0i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.32 - 2.29i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 2.08T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.26T + 53T^{2} \)
59 \( 1 + 10.6T + 59T^{2} \)
61 \( 1 + (-1.64 - 2.85i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (-2.45 + 4.25i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.36 + 2.36i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.52 + 13.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.84 - 11.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 4.46T + 89T^{2} \)
97 \( 1 + (-0.453 - 0.785i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.57530611764563469796914205761, −9.213730616090377159860394683178, −8.945396132252271892768526881727, −7.928938047395548900657465461466, −7.20974874404960822859342272121, −6.21421951841233261037573856398, −5.11154929521670251157667995002, −3.89151888991004303710072380311, −3.19126471747173875784542296364, −1.62714314150999692156941851691, 1.74105711486051530390101199662, 2.55813269034314397415309353503, 3.57488234180378721836734137404, 4.75828445548160373779705997640, 5.72762399720329472548134208312, 6.91335280179233561571013054284, 8.155780650493176364642335100581, 8.795925077056036450036586350811, 9.504059469615294242121989836594, 10.29738865637203166311409000424

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