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Properties

Label 2-1155-385.384-c1-0-44
Degree $2$
Conductor $1155$
Sign $0.292 - 0.956i$
Analytic cond. $9.22272$
Root an. cond. $3.03689$
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  + 1.83·2-s + 3-s + 1.35·4-s + (0.538 + 2.17i)5-s + 1.83·6-s + (0.996 + 2.45i)7-s − 1.17·8-s + 9-s + (0.986 + 3.97i)10-s + (−2.65 + 1.99i)11-s + 1.35·12-s + 1.25i·13-s + (1.82 + 4.49i)14-s + (0.538 + 2.17i)15-s − 4.87·16-s − 0.900i·17-s + ⋯
L(s)  = 1  + 1.29·2-s + 0.577·3-s + 0.679·4-s + (0.240 + 0.970i)5-s + 0.748·6-s + (0.376 + 0.926i)7-s − 0.415·8-s + 0.333·9-s + (0.312 + 1.25i)10-s + (−0.799 + 0.600i)11-s + 0.392·12-s + 0.347i·13-s + (0.488 + 1.20i)14-s + (0.139 + 0.560i)15-s − 1.21·16-s − 0.218i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.292 - 0.956i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.292 - 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.677385971\)
\(L(\frac12)\) \(\approx\) \(3.677385971\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + (-0.538 - 2.17i)T \)
7 \( 1 + (-0.996 - 2.45i)T \)
11 \( 1 + (2.65 - 1.99i)T \)
good2 \( 1 - 1.83T + 2T^{2} \)
13 \( 1 - 1.25iT - 13T^{2} \)
17 \( 1 + 0.900iT - 17T^{2} \)
19 \( 1 - 4.96T + 19T^{2} \)
23 \( 1 + 0.648iT - 23T^{2} \)
29 \( 1 + 8.58iT - 29T^{2} \)
31 \( 1 - 0.163iT - 31T^{2} \)
37 \( 1 - 5.80iT - 37T^{2} \)
41 \( 1 - 7.60T + 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 - 1.99T + 47T^{2} \)
53 \( 1 + 12.1iT - 53T^{2} \)
59 \( 1 - 0.600iT - 59T^{2} \)
61 \( 1 - 5.24T + 61T^{2} \)
67 \( 1 + 6.57iT - 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 - 1.42iT - 73T^{2} \)
79 \( 1 + 6.04iT - 79T^{2} \)
83 \( 1 + 6.80iT - 83T^{2} \)
89 \( 1 - 13.9iT - 89T^{2} \)
97 \( 1 + 19.2T + 97T^{2} \)
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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

−9.809353880687294601543300856381, −9.346279357857886198869767279315, −8.147983003889418788799283158386, −7.37395512006480024970505361562, −6.39194669045490334320944863400, −5.59923104049995582843964845717, −4.80545939659255282801619691907, −3.78043515326187900532361176290, −2.71749455719123626158403648885, −2.26913600845285550101456819003, 1.01790312379156594381006928267, 2.59607756333947686305534643593, 3.62203628675550291780365179715, 4.37889040087546884955480470176, 5.27640536270432626053081587225, 5.79573746397390850206188786265, 7.17648641411703326024038601210, 7.894609360681196425454995830620, 8.813983176203289920270259246062, 9.508114789657766123823737432195

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