Properties

Label 2-770-55.32-c1-0-7
Degree $2$
Conductor $770$
Sign $0.343 - 0.939i$
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 − 1.00i·4-s + (−2 − i)5-s + (0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + 3i·9-s + (2.12 − 0.707i)10-s + (−3 − 1.41i)11-s + (1.41 + 1.41i)13-s + 1.00i·14-s − 1.00·16-s + (−2.12 − 2.12i)18-s + 4.24·19-s + (−1.00 + 2.00i)20-s + (3.12 − 1.12i)22-s + (−2 + 2i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.894 − 0.447i)5-s + (0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s + i·9-s + (0.670 − 0.223i)10-s + (−0.904 − 0.426i)11-s + (0.392 + 0.392i)13-s + 0.267i·14-s − 0.250·16-s + (−0.499 − 0.499i)18-s + 0.973·19-s + (−0.223 + 0.447i)20-s + (0.665 − 0.239i)22-s + (−0.417 + 0.417i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.343 - 0.939i$
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: \((2,\ 770,\ (\ :1/2),\ 0.343 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.731652 + 0.511481i\)
\(L(\frac12)\) \(\approx\) \(0.731652 + 0.511481i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (3 + 1.41i)T \)
good3 \( 1 - 3iT^{2} \)
13 \( 1 + (-1.41 - 1.41i)T + 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 - 4.24T + 19T^{2} \)
23 \( 1 + (2 - 2i)T - 23iT^{2} \)
29 \( 1 - 7.07T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (-6 - 6i)T + 37iT^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 + (-5.65 - 5.65i)T + 43iT^{2} \)
47 \( 1 + (-7 - 7i)T + 47iT^{2} \)
53 \( 1 + (4 - 4i)T - 53iT^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 - 2.82iT - 61T^{2} \)
67 \( 1 + (3 + 3i)T + 67iT^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-4.24 - 4.24i)T + 73iT^{2} \)
79 \( 1 + 4.24T + 79T^{2} \)
83 \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \)
89 \( 1 + 14iT - 89T^{2} \)
97 \( 1 + (-6 - 6i)T + 97iT^{2} \)
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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

−10.52725588954706209720894518679, −9.527878312529593958592503526275, −8.508103623705158429010723981047, −7.86978718722745349747725999861, −7.43897779080180237454815393021, −6.11739477291568467300617191012, −5.08163076776503735119210157954, −4.35821044721616690263370439198, −2.86556166036080567521721632173, −1.13953218114641707511722086791, 0.65530401066015364356566676435, 2.50947526449916942830668017152, 3.44797687424685792106139419462, 4.44463948372988282171460184951, 5.75330273446599044718304449550, 6.93426699676010219117513401201, 7.71112599086515889039053184778, 8.442971184543783040033195167214, 9.325559132272720668137035667683, 10.31005969221271948082869424983

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