↑

Properties

Label 2-690-115.68-c1-0-8
Degree $2$
Conductor $690$
Sign $-0.0410 - 0.999i$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
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  + (−0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s − 1.00i·4-s + (2.13 + 0.677i)5-s − 1.00·6-s + (1.05 + 1.05i)7-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (−1.98 + 1.02i)10-s + 2.05i·11-s + (0.707 − 0.707i)12-s + (2.75 + 2.75i)13-s − 1.49·14-s + (1.02 + 1.98i)15-s − 1.00·16-s + (−4.76 − 4.76i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.408 + 0.408i)3-s − 0.500i·4-s + (0.952 + 0.303i)5-s − 0.408·6-s + (0.400 + 0.400i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (−0.628 + 0.324i)10-s + 0.619i·11-s + (0.204 − 0.204i)12-s + (0.763 + 0.763i)13-s − 0.400·14-s + (0.265 + 0.512i)15-s − 0.250·16-s + (−1.15 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0410 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0410 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.0410 - 0.999i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.0410 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12638 + 1.17361i\)
\(L(\frac12)\) \(\approx\) \(1.12638 + 1.17361i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-2.13 - 0.677i)T \)
23 \( 1 + (-2.97 + 3.76i)T \)
good7 \( 1 + (-1.05 - 1.05i)T + 7iT^{2} \)
11 \( 1 - 2.05iT - 11T^{2} \)
13 \( 1 + (-2.75 - 2.75i)T + 13iT^{2} \)
17 \( 1 + (4.76 + 4.76i)T + 17iT^{2} \)
19 \( 1 + 1.85T + 19T^{2} \)
29 \( 1 - 7.96iT - 29T^{2} \)
31 \( 1 - 10.3T + 31T^{2} \)
37 \( 1 + (2.66 + 2.66i)T + 37iT^{2} \)
41 \( 1 + 8.60T + 41T^{2} \)
43 \( 1 + (1.45 - 1.45i)T - 43iT^{2} \)
47 \( 1 + (-1.89 + 1.89i)T - 47iT^{2} \)
53 \( 1 + (3.57 - 3.57i)T - 53iT^{2} \)
59 \( 1 - 3.82iT - 59T^{2} \)
61 \( 1 - 2.76iT - 61T^{2} \)
67 \( 1 + (7.21 + 7.21i)T + 67iT^{2} \)
71 \( 1 - 0.620T + 71T^{2} \)
73 \( 1 + (-4.77 - 4.77i)T + 73iT^{2} \)
79 \( 1 - 16.3T + 79T^{2} \)
83 \( 1 + (5.71 - 5.71i)T - 83iT^{2} \)
89 \( 1 + 0.705T + 89T^{2} \)
97 \( 1 + (0.564 + 0.564i)T + 97iT^{2} \)
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

−10.53416549705073427997417626769, −9.629860627926566410180362715341, −8.931468469640460606439282164787, −8.431514952435396003455915488750, −6.95563253365095127127755764433, −6.55977106504729652918151638833, −5.23601461136601483444838530938, −4.51302695005839617129022795522, −2.78170422498581470921798753906, −1.73824825049559797912547312737, 1.06545709316309118480278854743, 2.13709103764508049979150005825, 3.35670165260349465768467588909, 4.59221691629500998147479575814, 5.96777504139310369697929959879, 6.67265349810997291177714952105, 8.138377312204192649364639804111, 8.398272231506960128185312201621, 9.358985832631715876164729192542, 10.30588785212913195458470282084

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