↑

Properties

Label 2-770-11.4-c1-0-21
Degree $2$
Conductor $770$
Sign $-0.809 + 0.587i$
Analytic cond. $6.14848$
Root an. cond. $2.47961$
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  + (−0.809 − 0.587i)2-s + (0.619 − 1.90i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (−1.62 + 1.17i)6-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.822 − 0.597i)9-s − 10-s + (−0.948 − 3.17i)11-s + 2.00·12-s + (3.25 + 2.36i)13-s + (−0.309 + 0.951i)14-s + (−0.619 − 1.90i)15-s + (−0.809 + 0.587i)16-s + (−0.243 + 0.176i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.357 − 1.10i)3-s + (0.154 + 0.475i)4-s + (0.361 − 0.262i)5-s + (−0.661 + 0.480i)6-s + (−0.116 − 0.359i)7-s + (0.109 − 0.336i)8-s + (−0.274 − 0.199i)9-s − 0.316·10-s + (−0.286 − 0.958i)11-s + 0.578·12-s + (0.904 + 0.656i)13-s + (−0.0825 + 0.254i)14-s + (−0.159 − 0.492i)15-s + (−0.202 + 0.146i)16-s + (−0.0589 + 0.0428i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.809 + 0.587i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.809 + 0.587i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.402286 - 1.23863i\)
\(L(\frac12)\) \(\approx\) \(0.402286 - 1.23863i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (0.948 + 3.17i)T \)
good3 \( 1 + (-0.619 + 1.90i)T + (-2.42 - 1.76i)T^{2} \)
13 \( 1 + (-3.25 - 2.36i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.243 - 0.176i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.39 + 7.35i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 6.48T + 23T^{2} \)
29 \( 1 + (-0.561 - 1.72i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.09 + 4.42i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.81 - 8.66i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.99 + 9.20i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.42T + 43T^{2} \)
47 \( 1 + (1.91 - 5.90i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.17 + 3.75i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.70 - 5.24i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.93 + 2.13i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 0.505T + 67T^{2} \)
71 \( 1 + (6.77 - 4.92i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.433 - 1.33i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.01 + 3.64i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (11.7 - 8.56i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 1.92T + 89T^{2} \)
97 \( 1 + (-13.3 - 9.69i)T + (29.9 + 92.2i)T^{2} \)
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

−9.878514688589378463501881497692, −8.982961052036583511881595397223, −8.351759209042892253491436879388, −7.49675537191927205287839485415, −6.70212787468727620141464813821, −5.80334445982169470661930667875, −4.28938339332400154685202863685, −3.00732046145111405565290157190, −1.91292914718428032621642326260, −0.78701979824071185743574844746, 1.81212717959815605660375564155, 3.26992728475042188186947490516, 4.25728543552935290314913210695, 5.49128327696805776271242329071, 6.13656970202285742785233088248, 7.41522112645548335758074779064, 8.197357304930399851837934281175, 9.116345455142288759304527987032, 9.888252597968814620634232791886, 10.22004195972388150084489198022

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