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Properties

Label 2-770-11.5-c1-0-20
Degree $2$
Conductor $770$
Sign $0.594 + 0.803i$
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  + (0.309 + 0.951i)2-s + (0.338 + 0.246i)3-s + (−0.809 + 0.587i)4-s + (0.309 − 0.951i)5-s + (−0.129 + 0.398i)6-s + (−0.809 + 0.587i)7-s + (−0.809 − 0.587i)8-s + (−0.872 − 2.68i)9-s + 0.999·10-s + (−1.19 − 3.09i)11-s − 0.418·12-s + (−1.63 − 5.04i)13-s + (−0.809 − 0.587i)14-s + (0.338 − 0.246i)15-s + (0.309 − 0.951i)16-s + (0.456 − 1.40i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.195 + 0.142i)3-s + (−0.404 + 0.293i)4-s + (0.138 − 0.425i)5-s + (−0.0528 + 0.162i)6-s + (−0.305 + 0.222i)7-s + (−0.286 − 0.207i)8-s + (−0.290 − 0.895i)9-s + 0.316·10-s + (−0.360 − 0.932i)11-s − 0.120·12-s + (−0.454 − 1.39i)13-s + (−0.216 − 0.157i)14-s + (0.0874 − 0.0635i)15-s + (0.0772 − 0.237i)16-s + (0.110 − 0.340i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 + 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.594 + 0.803i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.594 + 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10021 - 0.554663i\)
\(L(\frac12)\) \(\approx\) \(1.10021 - 0.554663i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (1.19 + 3.09i)T \)
good3 \( 1 + (-0.338 - 0.246i)T + (0.927 + 2.85i)T^{2} \)
13 \( 1 + (1.63 + 5.04i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.456 + 1.40i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.770 + 0.559i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 2.37T + 23T^{2} \)
29 \( 1 + (4.83 - 3.51i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.724 - 2.22i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.0181 - 0.0131i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.23 - 0.900i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 3.83T + 43T^{2} \)
47 \( 1 + (-1.21 - 0.884i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.50 + 10.7i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.622 - 0.451i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.49 + 7.66i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 + (3.68 - 11.3i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.89 - 4.28i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.95 + 12.1i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.89 + 5.83i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 + (-5.71 - 17.5i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.972897117670402749519136014136, −9.154811937695360289337633700436, −8.494747888534817477875022885645, −7.67318044119574982982819359694, −6.58499960251789066869117798845, −5.67816782692836655850960133947, −5.07609459469959131448836468172, −3.63153496246401045740698913083, −2.86097757212948843759443269885, −0.55019195060928819677810782733, 1.87388750348661801691305922687, 2.61959398039217536827517766879, 3.99414584934281600401125906800, 4.83007299517943746906457132752, 5.96843320397857249145709897066, 7.08097558424469317311506282017, 7.77252071670877522675516595869, 9.003133254675079970703006048614, 9.710142851775328154256768511103, 10.50925034345857304914105400351

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