Properties

Label 2-770-55.32-c1-0-11
Degree $2$
Conductor $770$
Sign $0.394 - 0.919i$
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  + (−0.707 + 0.707i)2-s + (−2.14 + 2.14i)3-s − 1.00i·4-s + (−1.71 + 1.43i)5-s − 3.04i·6-s + (0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s − 6.24i·9-s + (0.198 − 2.22i)10-s + (0.280 − 3.30i)11-s + (2.14 + 2.14i)12-s + (3.30 + 3.30i)13-s + 1.00i·14-s + (0.602 − 6.77i)15-s − 1.00·16-s + (2.29 − 2.29i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−1.24 + 1.24i)3-s − 0.500i·4-s + (−0.766 + 0.641i)5-s − 1.24i·6-s + (0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s − 2.08i·9-s + (0.0626 − 0.704i)10-s + (0.0844 − 0.996i)11-s + (0.620 + 0.620i)12-s + (0.916 + 0.916i)13-s + 0.267i·14-s + (0.155 − 1.74i)15-s − 0.250·16-s + (0.556 − 0.556i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.394 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.394 - 0.919i$
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: \((2,\ 770,\ (\ :1/2),\ 0.394 - 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.534807 + 0.352567i\)
\(L(\frac12)\) \(\approx\) \(0.534807 + 0.352567i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 - 1.43i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (-0.280 + 3.30i)T \)
good3 \( 1 + (2.14 - 2.14i)T - 3iT^{2} \)
13 \( 1 + (-3.30 - 3.30i)T + 13iT^{2} \)
17 \( 1 + (-2.29 + 2.29i)T - 17iT^{2} \)
19 \( 1 + 0.396T + 19T^{2} \)
23 \( 1 + (-5.93 + 5.93i)T - 23iT^{2} \)
29 \( 1 + 4.95T + 29T^{2} \)
31 \( 1 - 3.24T + 31T^{2} \)
37 \( 1 + (5.43 + 5.43i)T + 37iT^{2} \)
41 \( 1 - 6.00iT - 41T^{2} \)
43 \( 1 + (-7.83 - 7.83i)T + 43iT^{2} \)
47 \( 1 + (3.29 + 3.29i)T + 47iT^{2} \)
53 \( 1 + (9.91 - 9.91i)T - 53iT^{2} \)
59 \( 1 + 12.4iT - 59T^{2} \)
61 \( 1 + 7.90iT - 61T^{2} \)
67 \( 1 + (-8.03 - 8.03i)T + 67iT^{2} \)
71 \( 1 - 6.66T + 71T^{2} \)
73 \( 1 + (-3.89 - 3.89i)T + 73iT^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 + (1.89 + 1.89i)T + 83iT^{2} \)
89 \( 1 + 5.86iT - 89T^{2} \)
97 \( 1 + (-3.24 - 3.24i)T + 97iT^{2} \)
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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

−10.74644887713609375707785263009, −9.674666256888731258287001814153, −8.910502309872944563130280048384, −7.935837207754861673063388193150, −6.71890959741784470246560734743, −6.22367395556436196449359469656, −5.12153745713889956415881945862, −4.26391292702108789425826654629, −3.32144557407222704158068452920, −0.68279495006822838197642367626, 0.893870128067532688617839367952, 1.78385870687173394251668565406, 3.56082845652816357841013099072, 4.97040121033632905131143889818, 5.67551590579653433722520322147, 6.89343173311861280978250306858, 7.63088018976214612353807945830, 8.225629803576100742505327932376, 9.259667298189681922516231893925, 10.52368842200368465385872568895

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