Properties

Label 2-770-385.157-c1-0-31
Degree $2$
Conductor $770$
Sign $0.649 - 0.760i$
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  + (−0.358 − 0.933i)2-s + (1.75 + 2.69i)3-s + (−0.743 + 0.669i)4-s + (2.03 + 0.932i)5-s + (1.89 − 2.60i)6-s + (2.64 + 0.165i)7-s + (0.891 + 0.453i)8-s + (−2.99 + 6.72i)9-s + (0.142 − 2.23i)10-s + (−1.08 − 3.13i)11-s + (−3.10 − 0.833i)12-s + (2.83 + 0.449i)13-s + (−0.792 − 2.52i)14-s + (1.04 + 7.12i)15-s + (0.104 − 0.994i)16-s + (1.82 − 4.74i)17-s + ⋯
L(s)  = 1  + (−0.253 − 0.660i)2-s + (1.01 + 1.55i)3-s + (−0.371 + 0.334i)4-s + (0.908 + 0.417i)5-s + (0.772 − 1.06i)6-s + (0.998 + 0.0624i)7-s + (0.315 + 0.160i)8-s + (−0.997 + 2.24i)9-s + (0.0451 − 0.705i)10-s + (−0.328 − 0.944i)11-s + (−0.897 − 0.240i)12-s + (0.787 + 0.124i)13-s + (−0.211 − 0.674i)14-s + (0.269 + 1.83i)15-s + (0.0261 − 0.248i)16-s + (0.441 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 - 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.649 - 0.760i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.649 - 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10901 + 0.972970i\)
\(L(\frac12)\) \(\approx\) \(2.10901 + 0.972970i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.358 + 0.933i)T \)
5 \( 1 + (-2.03 - 0.932i)T \)
7 \( 1 + (-2.64 - 0.165i)T \)
11 \( 1 + (1.08 + 3.13i)T \)
good3 \( 1 + (-1.75 - 2.69i)T + (-1.22 + 2.74i)T^{2} \)
13 \( 1 + (-2.83 - 0.449i)T + (12.3 + 4.01i)T^{2} \)
17 \( 1 + (-1.82 + 4.74i)T + (-12.6 - 11.3i)T^{2} \)
19 \( 1 + (-2.30 + 2.56i)T + (-1.98 - 18.8i)T^{2} \)
23 \( 1 + (-5.32 - 1.42i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (6.19 + 2.01i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (10.5 - 1.10i)T + (30.3 - 6.44i)T^{2} \)
37 \( 1 + (7.64 + 4.96i)T + (15.0 + 33.8i)T^{2} \)
41 \( 1 + (6.13 - 1.99i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (5.89 - 5.89i)T - 43iT^{2} \)
47 \( 1 + (0.189 + 3.62i)T + (-46.7 + 4.91i)T^{2} \)
53 \( 1 + (2.00 + 1.61i)T + (11.0 + 51.8i)T^{2} \)
59 \( 1 + (4.09 + 4.54i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (0.185 + 0.0194i)T + (59.6 + 12.6i)T^{2} \)
67 \( 1 + (-1.55 + 0.415i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.66 - 1.93i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.111 + 2.12i)T + (-72.6 - 7.63i)T^{2} \)
79 \( 1 + (-0.0795 + 0.178i)T + (-52.8 - 58.7i)T^{2} \)
83 \( 1 + (-0.0426 - 0.269i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (0.977 + 1.69i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.65 - 10.4i)T + (-92.2 - 29.9i)T^{2} \)
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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

−10.46228040853197041211947883458, −9.416440991736158469764979827518, −9.118666264621376492750093626820, −8.316350528501728587229264738239, −7.30692970289839646201869808045, −5.39246116149034707440070524894, −5.07015721698415795009775723962, −3.59600531443532750327022429271, −3.03988169252985741241844157718, −1.85070997777273175606688886441, 1.47370804031724556941927398646, 1.83913443318180385040932698360, 3.53071409366836065357748803910, 5.16853773259495387585654886569, 5.95266775995406392125213196998, 7.02741908461143543792153938688, 7.56762911209057604008571797379, 8.521106235179482546702604124144, 8.819457874194265041530712697567, 9.903258367106618706425324727573

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