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Properties

Label 2-770-385.142-c1-0-9
Degree $2$
Conductor $770$
Sign $-0.201 - 0.979i$
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  + (0.258 + 0.965i)2-s + (−1.88 − 0.504i)3-s + (−0.866 + 0.499i)4-s + (−1.86 + 1.23i)5-s − 1.94i·6-s + (1.82 − 1.91i)7-s + (−0.707 − 0.707i)8-s + (0.687 + 0.397i)9-s + (−1.67 − 1.48i)10-s + (2.83 − 1.72i)11-s + (1.88 − 0.504i)12-s + (−0.386 − 0.386i)13-s + (2.32 + 1.26i)14-s + (4.13 − 1.37i)15-s + (0.500 − 0.866i)16-s + (3.57 + 0.959i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−1.08 − 0.291i)3-s + (−0.433 + 0.249i)4-s + (−0.835 + 0.550i)5-s − 0.795i·6-s + (0.689 − 0.724i)7-s + (−0.249 − 0.249i)8-s + (0.229 + 0.132i)9-s + (−0.528 − 0.469i)10-s + (0.853 − 0.521i)11-s + (0.543 − 0.145i)12-s + (−0.107 − 0.107i)13-s + (0.621 + 0.338i)14-s + (1.06 − 0.354i)15-s + (0.125 − 0.216i)16-s + (0.868 + 0.232i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.201 - 0.979i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.201 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.522200 + 0.640242i\)
\(L(\frac12)\) \(\approx\) \(0.522200 + 0.640242i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 + (1.86 - 1.23i)T \)
7 \( 1 + (-1.82 + 1.91i)T \)
11 \( 1 + (-2.83 + 1.72i)T \)
good3 \( 1 + (1.88 + 0.504i)T + (2.59 + 1.5i)T^{2} \)
13 \( 1 + (0.386 + 0.386i)T + 13iT^{2} \)
17 \( 1 + (-3.57 - 0.959i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.63 - 6.29i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.63 - 6.11i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 5.59T + 29T^{2} \)
31 \( 1 + (1.53 + 2.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.87 - 0.501i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 7.02iT - 41T^{2} \)
43 \( 1 + (-5.80 - 5.80i)T + 43iT^{2} \)
47 \( 1 + (4.09 - 1.09i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-13.9 - 3.72i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.22 - 3.01i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.59 - 0.918i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.10 - 11.5i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 + (3.80 - 14.2i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.67 + 2.89i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.90 - 3.90i)T + 83iT^{2} \)
89 \( 1 + (-11.7 - 6.78i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.09 + 1.09i)T + 97iT^{2} \)
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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

−10.83794246681093640373475874283, −9.828649142916690073359165592630, −8.499412960881748176357216596482, −7.72355569597670508503601009051, −7.08362434920478680054524863171, −6.12888172350943694306517599229, −5.49391356019086464784407697170, −4.20705783065186703373672380892, −3.51901558570335126128908957222, −1.15140411731668157793616634178, 0.55688999797959053846309465563, 2.18272054013791549675299790963, 3.77980142302601273612147445986, 4.82327260342119323229877235527, 5.14457894757477574872955832580, 6.36647025967018101932416076335, 7.48565572648714934916999403890, 8.750175194018223055858668395888, 9.103876581010756267030466751757, 10.46563635513227360541851102852

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