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Properties

Label 2-770-7.2-c1-0-1
Degree $2$
Conductor $770$
Sign $0.968 - 0.250i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−1.5 − 2.59i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 3·6-s + (0.5 + 2.59i)7-s − 0.999·8-s + (−3 + 5.19i)9-s + (0.499 + 0.866i)10-s + (0.5 + 0.866i)11-s + (−1.50 + 2.59i)12-s − 13-s + (2.5 + 0.866i)14-s + 3·15-s + (−0.5 + 0.866i)16-s + (3 + 5.19i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.866 − 1.49i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s − 1.22·6-s + (0.188 + 0.981i)7-s − 0.353·8-s + (−1 + 1.73i)9-s + (0.158 + 0.273i)10-s + (0.150 + 0.261i)11-s + (−0.433 + 0.749i)12-s − 0.277·13-s + (0.668 + 0.231i)14-s + 0.774·15-s + (−0.125 + 0.216i)16-s + (0.727 + 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.968 - 0.250i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.770694 + 0.0982123i\)
\(L(\frac12)\) \(\approx\) \(0.770694 + 0.0982123i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 2.59i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (1.5 + 2.59i)T + (-1.5 + 2.59i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 12T + 41T^{2} \)
43 \( 1 - 12T + 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.5 + 12.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (-2 - 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.5 - 12.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.68509369245788636922503504893, −9.731271656970502917841003484603, −8.385710032186939401586819752993, −7.75431857670758182215765210008, −6.65741592876568133460353999188, −5.90664306893209255558221025352, −5.29311412963579145413654575965, −3.76971212295114195822151290766, −2.31707875690740896976058529165, −1.52312271654808424633042596618, 0.39855784047227777662569564939, 3.27436005740129424533256513330, 4.21030299699161782457113001161, 4.85373361550935552731129289593, 5.52571759681256844065166773979, 6.67751624607992742553295741407, 7.53297287493351635809974377377, 8.770269465631275574098684014165, 9.429117371247022500067917414245, 10.41736470315123678448901305615

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