↑

Properties

Label 2-690-345.344-c1-0-14
Degree $2$
Conductor $690$
Sign $0.325 - 0.945i$
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  + 2-s + (−1.44 + 0.951i)3-s + 4-s + (−1.42 + 1.71i)5-s + (−1.44 + 0.951i)6-s + 4.73·7-s + 8-s + (1.18 − 2.75i)9-s + (−1.42 + 1.71i)10-s + 0.109·11-s + (−1.44 + 0.951i)12-s − 2.46i·13-s + 4.73·14-s + (0.431 − 3.84i)15-s + 16-s + 3.72i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.835 + 0.549i)3-s + 0.5·4-s + (−0.639 + 0.768i)5-s + (−0.590 + 0.388i)6-s + 1.79·7-s + 0.353·8-s + (0.395 − 0.918i)9-s + (−0.452 + 0.543i)10-s + 0.0330·11-s + (−0.417 + 0.274i)12-s − 0.685i·13-s + 1.26·14-s + (0.111 − 0.993i)15-s + 0.250·16-s + 0.902i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.325 - 0.945i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.325 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54232 + 1.09961i\)
\(L(\frac12)\) \(\approx\) \(1.54232 + 1.09961i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1.44 - 0.951i)T \)
5 \( 1 + (1.42 - 1.71i)T \)
23 \( 1 + (-4.67 - 1.04i)T \)
good7 \( 1 - 4.73T + 7T^{2} \)
11 \( 1 - 0.109T + 11T^{2} \)
13 \( 1 + 2.46iT - 13T^{2} \)
17 \( 1 - 3.72iT - 17T^{2} \)
19 \( 1 - 6.76iT - 19T^{2} \)
29 \( 1 - 7.05iT - 29T^{2} \)
31 \( 1 + 9.34T + 31T^{2} \)
37 \( 1 - 1.34T + 37T^{2} \)
41 \( 1 + 7.38iT - 41T^{2} \)
43 \( 1 + 2.31T + 43T^{2} \)
47 \( 1 - 5.53T + 47T^{2} \)
53 \( 1 - 3.77iT - 53T^{2} \)
59 \( 1 + 4.18iT - 59T^{2} \)
61 \( 1 - 7.57iT - 61T^{2} \)
67 \( 1 - 5.95T + 67T^{2} \)
71 \( 1 + 9.66iT - 71T^{2} \)
73 \( 1 - 9.28iT - 73T^{2} \)
79 \( 1 + 12.3iT - 79T^{2} \)
83 \( 1 + 16.6iT - 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 - 9.94T + 97T^{2} \)
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

−10.66694878472398944950128991439, −10.48888320693225531194169115162, −8.804989187452653759271946727462, −7.79190034042512961942115131649, −7.11116030084506712394847472772, −5.83274229144644918994820250722, −5.22586937481586626464957889299, −4.19255633227446558162999607280, −3.43416514294166648267177802672, −1.61952515625249064205159801130, 1.00804797099976319913492739226, 2.26278585465881841639006796700, 4.24727107439866823732456246803, 4.88848177765801972883790486129, 5.40059264746194706993922847276, 6.84768567692439589281584675312, 7.49853088325972260431411613907, 8.318806807360994224763962963549, 9.339310806659410749655383579891, 10.99646775982930852106378593808

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