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Properties

Label 2-770-11.4-c1-0-12
Degree $2$
Conductor $770$
Sign $-0.394 - 0.918i$
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  + (0.809 + 0.587i)2-s + (−0.5 + 1.53i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (−1.30 + 0.951i)6-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (0.309 + 0.224i)9-s + 10-s + (3.30 − 0.224i)11-s − 1.61·12-s + (0.618 + 0.449i)13-s + (−0.309 + 0.951i)14-s + (0.5 + 1.53i)15-s + (−0.809 + 0.587i)16-s + (−0.5 + 0.363i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.288 + 0.888i)3-s + (0.154 + 0.475i)4-s + (0.361 − 0.262i)5-s + (−0.534 + 0.388i)6-s + (0.116 + 0.359i)7-s + (−0.109 + 0.336i)8-s + (0.103 + 0.0748i)9-s + 0.316·10-s + (0.997 − 0.0676i)11-s − 0.467·12-s + (0.171 + 0.124i)13-s + (−0.0825 + 0.254i)14-s + (0.129 + 0.397i)15-s + (−0.202 + 0.146i)16-s + (−0.121 + 0.0881i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.394 - 0.918i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.394 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18254 + 1.79491i\)
\(L(\frac12)\) \(\approx\) \(1.18254 + 1.79491i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-3.30 + 0.224i)T \)
good3 \( 1 + (0.5 - 1.53i)T + (-2.42 - 1.76i)T^{2} \)
13 \( 1 + (-0.618 - 0.449i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.5 - 0.363i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.809 - 2.48i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 0.472T + 23T^{2} \)
29 \( 1 + (-1.23 - 3.80i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.61 + 2.62i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.14 + 3.52i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.97 + 6.06i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.38T + 43T^{2} \)
47 \( 1 + (3.47 - 10.6i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2 + 1.45i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.427 + 1.31i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-6.47 + 4.70i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 5.09T + 67T^{2} \)
71 \( 1 + (-5.23 + 3.80i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.5 + 10.7i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-9.47 - 6.88i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.0729 + 0.0530i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 3.14T + 89T^{2} \)
97 \( 1 + (11.1 + 8.11i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.68916169025125010008904971460, −9.602578255529839268349884895186, −9.044432177788156487664542954730, −8.027325502275114945087588074203, −6.89252203225853931192566563304, −5.98205391920920461629917085968, −5.19562047942409531131931030638, −4.32341869038854605450179696358, −3.50710872506245741197475264460, −1.84364340299893270352310087809, 1.03324414962351971240185390136, 2.11124974885591595399890502008, 3.49503906684165804982781291450, 4.51966688842777777231854246349, 5.70285091538498267135874672427, 6.66389345787293434210768230083, 7.00049301573007638728868220832, 8.265147087890703267371056077997, 9.405250132371369006069010562452, 10.14385320701297954344442956655

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