Properties

Label 2-690-3.2-c2-0-35
Degree $2$
Conductor $690$
Sign $0.955 - 0.293i$
Analytic cond. $18.8011$
Root an. cond. $4.33602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s + (0.880 + 2.86i)3-s − 2.00·4-s − 2.23i·5-s + (−4.05 + 1.24i)6-s − 3.20·7-s − 2.82i·8-s + (−7.45 + 5.04i)9-s + 3.16·10-s − 14.6i·11-s + (−1.76 − 5.73i)12-s + 18.4·13-s − 4.53i·14-s + (6.41 − 1.96i)15-s + 4.00·16-s − 27.0i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.293 + 0.955i)3-s − 0.500·4-s − 0.447i·5-s + (−0.675 + 0.207i)6-s − 0.458·7-s − 0.353i·8-s + (−0.827 + 0.560i)9-s + 0.316·10-s − 1.33i·11-s + (−0.146 − 0.477i)12-s + 1.42·13-s − 0.324i·14-s + (0.427 − 0.131i)15-s + 0.250·16-s − 1.59i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.955 - 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.955 - 0.293i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.955 - 0.293i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.610131998\)
\(L(\frac12)\) \(\approx\) \(1.610131998\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + (-0.880 - 2.86i)T \)
5 \( 1 + 2.23iT \)
23 \( 1 - 4.79iT \)
good7 \( 1 + 3.20T + 49T^{2} \)
11 \( 1 + 14.6iT - 121T^{2} \)
13 \( 1 - 18.4T + 169T^{2} \)
17 \( 1 + 27.0iT - 289T^{2} \)
19 \( 1 + 5.52T + 361T^{2} \)
29 \( 1 - 8.45iT - 841T^{2} \)
31 \( 1 - 49.6T + 961T^{2} \)
37 \( 1 - 42.6T + 1.36e3T^{2} \)
41 \( 1 + 20.1iT - 1.68e3T^{2} \)
43 \( 1 + 17.0T + 1.84e3T^{2} \)
47 \( 1 + 52.7iT - 2.20e3T^{2} \)
53 \( 1 + 3.72iT - 2.80e3T^{2} \)
59 \( 1 + 29.1iT - 3.48e3T^{2} \)
61 \( 1 - 8.31T + 3.72e3T^{2} \)
67 \( 1 - 0.327T + 4.48e3T^{2} \)
71 \( 1 - 69.2iT - 5.04e3T^{2} \)
73 \( 1 - 22.6T + 5.32e3T^{2} \)
79 \( 1 - 68.6T + 6.24e3T^{2} \)
83 \( 1 + 73.2iT - 6.88e3T^{2} \)
89 \( 1 - 72.4iT - 7.92e3T^{2} \)
97 \( 1 - 89.0T + 9.40e3T^{2} \)
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

−10.07830903342316276995302680239, −9.258836228131114680322243301736, −8.609622048538265009520201586520, −8.006103844582549892301910135548, −6.58012015963676190855370581284, −5.77529110475259860920322425415, −4.91053904525633754491337270400, −3.82944765738678753676265489017, −2.96751128241928780725208239857, −0.63040296899054402283664669207, 1.24711355156841704535246616179, 2.28746356600113825664038036300, 3.38875556588473076816426858699, 4.39428984952113365869410604921, 6.11465846452268642656451145521, 6.50628902856849626326203749088, 7.79507858756216572213591409195, 8.423873108709377560605322941242, 9.450882714824862194699067802016, 10.29358222218680451636444203704

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