↑

Properties

Label 2-670-67.64-c1-0-19
Degree $2$
Conductor $670$
Sign $-0.473 - 0.880i$
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  + (−0.959 − 0.281i)2-s + (0.396 − 2.76i)3-s + (0.841 + 0.540i)4-s + (−0.415 − 0.909i)5-s + (−1.15 + 2.53i)6-s + (−2.99 − 0.880i)7-s + (−0.654 − 0.755i)8-s + (−4.58 − 1.34i)9-s + (0.142 + 0.989i)10-s + (0.459 + 1.00i)11-s + (1.82 − 2.10i)12-s + (0.511 − 0.590i)13-s + (2.62 + 1.68i)14-s + (−2.67 + 0.785i)15-s + (0.415 + 0.909i)16-s + (−1.75 + 1.12i)17-s + ⋯
L(s)  = 1  + (−0.678 − 0.199i)2-s + (0.229 − 1.59i)3-s + (0.420 + 0.270i)4-s + (−0.185 − 0.406i)5-s + (−0.473 + 1.03i)6-s + (−1.13 − 0.332i)7-s + (−0.231 − 0.267i)8-s + (−1.52 − 0.448i)9-s + (0.0450 + 0.313i)10-s + (0.138 + 0.303i)11-s + (0.527 − 0.608i)12-s + (0.141 − 0.163i)13-s + (0.702 + 0.451i)14-s + (−0.691 + 0.202i)15-s + (0.103 + 0.227i)16-s + (−0.424 + 0.272i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $-0.473 - 0.880i$
Analytic conductor: \(5.34997\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{670} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ -0.473 - 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.190572 + 0.318696i\)
\(L(\frac12)\) \(\approx\) \(0.190572 + 0.318696i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.959 + 0.281i)T \)
5 \( 1 + (0.415 + 0.909i)T \)
67 \( 1 + (5.35 + 6.19i)T \)
good3 \( 1 + (-0.396 + 2.76i)T + (-2.87 - 0.845i)T^{2} \)
7 \( 1 + (2.99 + 0.880i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (-0.459 - 1.00i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (-0.511 + 0.590i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (1.75 - 1.12i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (3.98 - 1.17i)T + (15.9 - 10.2i)T^{2} \)
23 \( 1 + (-0.193 + 1.34i)T + (-22.0 - 6.47i)T^{2} \)
29 \( 1 + 5.42T + 29T^{2} \)
31 \( 1 + (-2.25 - 2.59i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 - 3.28T + 37T^{2} \)
41 \( 1 + (-1.34 + 0.864i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (0.880 - 0.565i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 + (-1.69 + 11.7i)T + (-45.0 - 13.2i)T^{2} \)
53 \( 1 + (-2.68 - 1.72i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (2.75 + 3.17i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (5.12 - 11.2i)T + (-39.9 - 46.1i)T^{2} \)
71 \( 1 + (-7.51 - 4.83i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (4.41 - 9.66i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (3.43 - 3.95i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (3.23 + 7.09i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.784 + 5.45i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
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

−9.829281240825421603963980439077, −8.842829498641608677978419617934, −8.225255237735709358634948787085, −7.25300550350916377779623034736, −6.70118235068816000768454416649, −5.86200668441838071596165787416, −4.00977687210639650912726947531, −2.73427195388099926624450426701, −1.59759388461227810593454087102, −0.23096096718435029156381887581, 2.60982477571466461919023124064, 3.53674653363917844900972756141, 4.53301427402758593653065471044, 5.84108182491586047310558308854, 6.57074093126435361792079331169, 7.81028914086011308845960916317, 8.934638329916148802784522526869, 9.340190736925903852884824287742, 10.03521465152377466938188757829, 10.88188798699265248929786438403

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