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Properties

Label 2-770-11.9-c1-0-14
Degree $2$
Conductor $770$
Sign $0.569 + 0.821i$
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 + (−1.30 + 0.951i)3-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + (0.499 + 1.53i)6-s + (−0.809 − 0.587i)7-s + (−0.809 + 0.587i)8-s + (−0.118 + 0.363i)9-s + 0.999·10-s + (−2.54 − 2.12i)11-s + 1.61·12-s + (1.73 − 5.34i)13-s + (−0.809 + 0.587i)14-s + (−1.30 − 0.951i)15-s + (0.309 + 0.951i)16-s + (1.42 + 4.39i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.755 + 0.549i)3-s + (−0.404 − 0.293i)4-s + (0.138 + 0.425i)5-s + (0.204 + 0.628i)6-s + (−0.305 − 0.222i)7-s + (−0.286 + 0.207i)8-s + (−0.0393 + 0.121i)9-s + 0.316·10-s + (−0.767 − 0.641i)11-s + 0.467·12-s + (0.481 − 1.48i)13-s + (−0.216 + 0.157i)14-s + (−0.337 − 0.245i)15-s + (0.0772 + 0.237i)16-s + (0.346 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.569 + 0.821i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.569 + 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03218 - 0.540296i\)
\(L(\frac12)\) \(\approx\) \(1.03218 - 0.540296i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.54 + 2.12i)T \)
good3 \( 1 + (1.30 - 0.951i)T + (0.927 - 2.85i)T^{2} \)
13 \( 1 + (-1.73 + 5.34i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.42 - 4.39i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.23 + 3.07i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 8.47T + 23T^{2} \)
29 \( 1 + (-4.73 - 3.44i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.61 + 4.97i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3 - 2.17i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (7.23 - 5.25i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 3.23T + 43T^{2} \)
47 \( 1 + (-10.5 + 7.69i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2 + 6.15i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.85 + 2.80i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.61 + 11.1i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 1.23T + 67T^{2} \)
71 \( 1 + (2.26 + 6.96i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (3.5 + 2.54i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.263 - 0.812i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.20 - 12.9i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 0.291T + 89T^{2} \)
97 \( 1 + (1.04 - 3.21i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.52200667552024866867514462467, −9.775233009457016601470223575683, −8.566443641444402067917852360279, −7.70206547448464295341124855829, −6.36414136328759875192700857919, −5.49164408372385082286634472231, −4.92370869163635007077076205438, −3.47703720975018988266166824690, −2.79402824520010078336075366722, −0.76671932625391353934574609807, 1.12139018908485059017963703793, 2.91837619627508184093047525697, 4.40528559429978115272105001685, 5.29217576853524540742451375631, 6.02724495504129424609876595371, 7.00294112227483189631005288861, 7.45092543402976955214561514779, 8.866744021311061507559988577675, 9.310078419330046424244612659691, 10.42205620896755638166334519588

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