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Properties

Label 2-770-55.43-c1-0-11
Degree $2$
Conductor $770$
Sign $-0.112 - 0.993i$
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 + 2i)3-s + 1.00i·4-s + (−1 + 2i)5-s − 2.82i·6-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + 5i·9-s + (2.12 − 0.707i)10-s + (3 + 1.41i)11-s + (−2.00 + 2.00i)12-s + (1.41 − 1.41i)13-s − 1.00i·14-s + (−6 + 2i)15-s − 1.00·16-s + (1.41 + 1.41i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (1.15 + 1.15i)3-s + 0.500i·4-s + (−0.447 + 0.894i)5-s − 1.15i·6-s + (0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s + 1.66i·9-s + (0.670 − 0.223i)10-s + (0.904 + 0.426i)11-s + (−0.577 + 0.577i)12-s + (0.392 − 0.392i)13-s − 0.267i·14-s + (−1.54 + 0.516i)15-s − 0.250·16-s + (0.342 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.112 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.112 - 0.993i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.112 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11077 + 1.24403i\)
\(L(\frac12)\) \(\approx\) \(1.11077 + 1.24403i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 2i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (-3 - 1.41i)T \)
good3 \( 1 + (-2 - 2i)T + 3iT^{2} \)
13 \( 1 + (-1.41 + 1.41i)T - 13iT^{2} \)
17 \( 1 + (-1.41 - 1.41i)T + 17iT^{2} \)
19 \( 1 + 4.24T + 19T^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (4 - 4i)T - 37iT^{2} \)
41 \( 1 + 4.24iT - 41T^{2} \)
43 \( 1 + (5.65 - 5.65i)T - 43iT^{2} \)
47 \( 1 + (-5 + 5i)T - 47iT^{2} \)
53 \( 1 + (6 + 6i)T + 53iT^{2} \)
59 \( 1 - 8iT - 59T^{2} \)
61 \( 1 + 5.65iT - 61T^{2} \)
67 \( 1 + (-3 + 3i)T - 67iT^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (11.3 - 11.3i)T - 73iT^{2} \)
79 \( 1 + 7.07T + 79T^{2} \)
83 \( 1 + (-7.07 + 7.07i)T - 83iT^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 + (-2 + 2i)T - 97iT^{2} \)
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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

−10.24944181527302607158737099850, −9.912457143646903550268571420203, −8.750387202231119101101219894662, −8.417805529027993643278282132882, −7.42228017478961630488691247979, −6.31500948553918222672058831076, −4.66292806403760146006033867634, −3.80020024167252191060552306712, −3.12792976410519124061027458445, −2.01403252375171773576052575104, 0.931388580131252379563727441700, 1.89627815102061108570217478180, 3.48389808131150981733814786214, 4.56883112588492768511340834689, 6.02128331918261053954021301463, 6.90199617582853896403059175197, 7.65333271089563636297060749301, 8.466995126943970353516802716366, 8.792752550756768190418024182608, 9.609565912814369728433387509154

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