Properties

Label 2-770-55.32-c1-0-0
Degree $2$
Conductor $770$
Sign $-0.505 - 0.862i$
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  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−2 − i)5-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s + 3i·9-s + (−2.12 + 0.707i)10-s + (−3 + 1.41i)11-s + (−1.41 − 1.41i)13-s + 1.00i·14-s − 1.00·16-s + (2.12 + 2.12i)18-s − 4.24·19-s + (−1.00 + 2.00i)20-s + (−1.12 + 3.12i)22-s + (−2 + 2i)23-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.894 − 0.447i)5-s + (−0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s + i·9-s + (−0.670 + 0.223i)10-s + (−0.904 + 0.426i)11-s + (−0.392 − 0.392i)13-s + 0.267i·14-s − 0.250·16-s + (0.499 + 0.499i)18-s − 0.973·19-s + (−0.223 + 0.447i)20-s + (−0.239 + 0.665i)22-s + (−0.417 + 0.417i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 - 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.505 - 0.862i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.505 - 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.120887 + 0.211047i\)
\(L(\frac12)\) \(\approx\) \(0.120887 + 0.211047i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (2 + i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (3 - 1.41i)T \)
good3 \( 1 - 3iT^{2} \)
13 \( 1 + (1.41 + 1.41i)T + 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 + 4.24T + 19T^{2} \)
23 \( 1 + (2 - 2i)T - 23iT^{2} \)
29 \( 1 + 7.07T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (-6 - 6i)T + 37iT^{2} \)
41 \( 1 + 1.41iT - 41T^{2} \)
43 \( 1 + (5.65 + 5.65i)T + 43iT^{2} \)
47 \( 1 + (-7 - 7i)T + 47iT^{2} \)
53 \( 1 + (4 - 4i)T - 53iT^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 + 2.82iT - 61T^{2} \)
67 \( 1 + (3 + 3i)T + 67iT^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (4.24 + 4.24i)T + 73iT^{2} \)
79 \( 1 - 4.24T + 79T^{2} \)
83 \( 1 + (8.48 + 8.48i)T + 83iT^{2} \)
89 \( 1 + 14iT - 89T^{2} \)
97 \( 1 + (-6 - 6i)T + 97iT^{2} \)
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

−10.70636192272774463861996515304, −9.984733488086228674215942383984, −8.910866982375228113329095162754, −7.933163001809871000054188056912, −7.36111786212466359651046600034, −5.92360388728646491487265115320, −5.01294941567582544771760225709, −4.31939352958825311363591430287, −3.08699148705917981377178459384, −1.97923180520971605931022434698, 0.096926374457960697344707824981, 2.62603718125499673843228938150, 3.70040971250586905111789545760, 4.38540375158162429065075813470, 5.70878160755590499972270568385, 6.57874254139703300076035958178, 7.32285069290052777824897882601, 8.128916673809894908178743143185, 9.009430903636280116608488454723, 10.09513852022276277141180119601

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