Properties

Label 2-690-115.114-c2-0-27
Degree $2$
Conductor $690$
Sign $-0.415 + 0.909i$
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 about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s + 1.73i·3-s − 2.00·4-s + (−1.71 + 4.69i)5-s + 2.44·6-s − 12.9·7-s + 2.82i·8-s − 2.99·9-s + (6.63 + 2.43i)10-s + 18.3i·11-s − 3.46i·12-s − 7.60i·13-s + 18.2i·14-s + (−8.13 − 2.97i)15-s + 4.00·16-s + 19.9·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.500·4-s + (−0.343 + 0.939i)5-s + 0.408·6-s − 1.84·7-s + 0.353i·8-s − 0.333·9-s + (0.663 + 0.243i)10-s + 1.67i·11-s − 0.288i·12-s − 0.584i·13-s + 1.30i·14-s + (−0.542 − 0.198i)15-s + 0.250·16-s + 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.415 + 0.909i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ -0.415 + 0.909i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3517271709\)
\(L(\frac12)\) \(\approx\) \(0.3517271709\)
\(L(2)\) 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 \( 1 - 1.73iT \)
5 \( 1 + (1.71 - 4.69i)T \)
23 \( 1 + (22.9 + 1.79i)T \)
good7 \( 1 + 12.9T + 49T^{2} \)
11 \( 1 - 18.3iT - 121T^{2} \)
13 \( 1 + 7.60iT - 169T^{2} \)
17 \( 1 - 19.9T + 289T^{2} \)
19 \( 1 + 24.3iT - 361T^{2} \)
29 \( 1 - 32.2T + 841T^{2} \)
31 \( 1 - 3.18T + 961T^{2} \)
37 \( 1 - 11.0T + 1.36e3T^{2} \)
41 \( 1 + 60.4T + 1.68e3T^{2} \)
43 \( 1 + 50.3T + 1.84e3T^{2} \)
47 \( 1 + 79.4iT - 2.20e3T^{2} \)
53 \( 1 - 92.1T + 2.80e3T^{2} \)
59 \( 1 - 5.16T + 3.48e3T^{2} \)
61 \( 1 - 10.5iT - 3.72e3T^{2} \)
67 \( 1 - 72.2T + 4.48e3T^{2} \)
71 \( 1 + 8.94T + 5.04e3T^{2} \)
73 \( 1 + 70.6iT - 5.32e3T^{2} \)
79 \( 1 - 1.35iT - 6.24e3T^{2} \)
83 \( 1 + 135.T + 6.88e3T^{2} \)
89 \( 1 - 41.9iT - 7.92e3T^{2} \)
97 \( 1 + 165.T + 9.40e3T^{2} \)
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

−10.01412104715790053232435085539, −9.723429656443466946692107867510, −8.452879634843737357347310638975, −7.19588192482706602558093733659, −6.56345107327554217454347353125, −5.30339100235299389842164858724, −4.06047370343131040090561577085, −3.25299577645816535819768612720, −2.47438500944085299568907147584, −0.15163710272434693309132174726, 1.03097940173050573685467779579, 3.17913915949737733034217930787, 3.93200151670176277156240488921, 5.57308143715922002320989084629, 6.05598874675168844826372710714, 6.90100798368670934194051428647, 8.108862317007341318307830830086, 8.512513564767975126532628108090, 9.563416362856935565949062784208, 10.20012122408985479287112089068

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