↑

Properties

Label 2-770-11.4-c1-0-20
Degree $2$
Conductor $770$
Sign $-0.480 + 0.877i$
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 + (−0.347 + 1.06i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (0.909 − 0.660i)6-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (1.40 + 1.02i)9-s − 10-s + (−2.21 − 2.46i)11-s − 1.12·12-s + (−5.71 − 4.15i)13-s + (−0.309 + 0.951i)14-s + (0.347 + 1.06i)15-s + (−0.809 + 0.587i)16-s + (−2.78 + 2.02i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.200 + 0.617i)3-s + (0.154 + 0.475i)4-s + (0.361 − 0.262i)5-s + (0.371 − 0.269i)6-s + (−0.116 − 0.359i)7-s + (0.109 − 0.336i)8-s + (0.467 + 0.340i)9-s − 0.316·10-s + (−0.668 − 0.743i)11-s − 0.324·12-s + (−1.58 − 1.15i)13-s + (−0.0825 + 0.254i)14-s + (0.0897 + 0.276i)15-s + (−0.202 + 0.146i)16-s + (−0.675 + 0.491i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 + 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.480 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.480 + 0.877i$
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: \((2,\ 770,\ (\ :1/2),\ -0.480 + 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.325076 - 0.548588i\)
\(L(\frac12)\) \(\approx\) \(0.325076 - 0.548588i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 + 2.46i)T \)
good3 \( 1 + (0.347 - 1.06i)T + (-2.42 - 1.76i)T^{2} \)
13 \( 1 + (5.71 + 4.15i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.78 - 2.02i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.06 + 3.26i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 3.63T + 23T^{2} \)
29 \( 1 + (2.68 + 8.26i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.65 - 2.65i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.241 + 0.742i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.55 - 4.77i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 5.39T + 43T^{2} \)
47 \( 1 + (-3.03 + 9.34i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (9.19 + 6.68i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.25 + 10.0i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.369 - 0.268i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 + (-5.51 + 4.00i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.67 + 14.3i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-13.1 - 9.54i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (5.98 - 4.35i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 4.97T + 89T^{2} \)
97 \( 1 + (2.33 + 1.69i)T + (29.9 + 92.2i)T^{2} \)
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

−9.951172425509350728555345559650, −9.585698198429319527169132525773, −8.356173476249345976165217482205, −7.68448533322973801259442193293, −6.66038521573094407748529967438, −5.28623008144686035996967733306, −4.69442963351669332111801377631, −3.31968441942573625006152313052, −2.22523913709542776026533992556, −0.38203433488891065566875038475, 1.64862663543279525337608781503, 2.63397794976757440853323025854, 4.50689444811586550692182681881, 5.40804726897598688016626986114, 6.61145825188995814272027724036, 7.08311202310667820117951061478, 7.74369429258738278942549497117, 9.110307762270660091481673946790, 9.588832982802863264337692317364, 10.35652668515318927233677983570

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