Properties

Label 2-770-385.152-c1-0-16
Degree $2$
Conductor $770$
Sign $0.458 - 0.888i$
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  + (−0.0523 + 0.998i)2-s + (−0.366 − 0.955i)3-s + (−0.994 − 0.104i)4-s + (1.54 + 1.61i)5-s + (0.973 − 0.316i)6-s + (−2.56 + 0.645i)7-s + (0.156 − 0.987i)8-s + (1.45 − 1.30i)9-s + (−1.69 + 1.45i)10-s + (2.64 + 2.00i)11-s + (0.265 + 0.989i)12-s + (2.72 − 1.39i)13-s + (−0.510 − 2.59i)14-s + (0.982 − 2.06i)15-s + (0.978 + 0.207i)16-s + (−0.237 − 4.52i)17-s + ⋯
L(s)  = 1  + (−0.0370 + 0.706i)2-s + (−0.211 − 0.551i)3-s + (−0.497 − 0.0522i)4-s + (0.689 + 0.724i)5-s + (0.397 − 0.129i)6-s + (−0.969 + 0.244i)7-s + (0.0553 − 0.349i)8-s + (0.483 − 0.435i)9-s + (−0.536 + 0.460i)10-s + (0.797 + 0.603i)11-s + (0.0765 + 0.285i)12-s + (0.756 − 0.385i)13-s + (−0.136 − 0.693i)14-s + (0.253 − 0.534i)15-s + (0.244 + 0.0519i)16-s + (−0.0575 − 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.458 - 0.888i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.458 - 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26711 + 0.771807i\)
\(L(\frac12)\) \(\approx\) \(1.26711 + 0.771807i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0523 - 0.998i)T \)
5 \( 1 + (-1.54 - 1.61i)T \)
7 \( 1 + (2.56 - 0.645i)T \)
11 \( 1 + (-2.64 - 2.00i)T \)
good3 \( 1 + (0.366 + 0.955i)T + (-2.22 + 2.00i)T^{2} \)
13 \( 1 + (-2.72 + 1.39i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (0.237 + 4.52i)T + (-16.9 + 1.77i)T^{2} \)
19 \( 1 + (-0.380 - 3.62i)T + (-18.5 + 3.95i)T^{2} \)
23 \( 1 + (-0.339 - 1.26i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.64 - 3.64i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.82 - 8.57i)T + (-28.3 + 12.6i)T^{2} \)
37 \( 1 + (-8.61 - 3.30i)T + (27.4 + 24.7i)T^{2} \)
41 \( 1 + (-2.62 - 3.61i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (-8.98 + 8.98i)T - 43iT^{2} \)
47 \( 1 + (-4.93 - 3.99i)T + (9.77 + 45.9i)T^{2} \)
53 \( 1 + (1.60 - 2.46i)T + (-21.5 - 48.4i)T^{2} \)
59 \( 1 + (0.632 - 6.01i)T + (-57.7 - 12.2i)T^{2} \)
61 \( 1 + (-2.35 + 11.0i)T + (-55.7 - 24.8i)T^{2} \)
67 \( 1 + (0.120 - 0.451i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.216 - 0.666i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.87 - 3.94i)T + (15.1 - 71.4i)T^{2} \)
79 \( 1 + (8.69 - 7.83i)T + (8.25 - 78.5i)T^{2} \)
83 \( 1 + (1.69 - 3.33i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (-2.00 + 3.47i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.88 + 11.5i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.16536204473154564188893439415, −9.567144968943158782542382734690, −8.902236075450481954313437216736, −7.46444378769282508553622908254, −6.90987584822379523380072496842, −6.24444563070997290170130912832, −5.56946433958566168709987247001, −4.03883392864776355657236205990, −2.94572298138814245326215123923, −1.29594349609390690545885619616, 0.960020318296133820742644962293, 2.36847992940384130077973876553, 3.92877939039909834579623347071, 4.33135627294647983281198715315, 5.77306025154376909792162145856, 6.30921029628597525121568167199, 7.77392602800731926136244521377, 8.985082839006308469510704767471, 9.355895258230572236155439004372, 10.14974729236144031666401941195

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