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Properties

Label 2-1155-385.384-c1-0-89
Degree $2$
Conductor $1155$
Sign $0.892 + 0.450i$
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  + 2.52·2-s − 3-s + 4.39·4-s + (1.63 − 1.52i)5-s − 2.52·6-s + (2.48 + 0.905i)7-s + 6.06·8-s + 9-s + (4.13 − 3.86i)10-s + (−2.13 − 2.53i)11-s − 4.39·12-s − 5.27i·13-s + (6.28 + 2.29i)14-s + (−1.63 + 1.52i)15-s + 6.54·16-s + 5.11i·17-s + ⋯
L(s)  = 1  + 1.78·2-s − 0.577·3-s + 2.19·4-s + (0.730 − 0.682i)5-s − 1.03·6-s + (0.939 + 0.342i)7-s + 2.14·8-s + 0.333·9-s + (1.30 − 1.22i)10-s + (−0.645 − 0.764i)11-s − 1.26·12-s − 1.46i·13-s + (1.68 + 0.612i)14-s + (−0.421 + 0.394i)15-s + 1.63·16-s + 1.24i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.450i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 + 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.892 + 0.450i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.892 + 0.450i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.698315059\)
\(L(\frac12)\) \(\approx\) \(4.698315059\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 + 1.52i)T \)
7 \( 1 + (-2.48 - 0.905i)T \)
11 \( 1 + (2.13 + 2.53i)T \)
good2 \( 1 - 2.52T + 2T^{2} \)
13 \( 1 + 5.27iT - 13T^{2} \)
17 \( 1 - 5.11iT - 17T^{2} \)
19 \( 1 + 4.28T + 19T^{2} \)
23 \( 1 - 1.22iT - 23T^{2} \)
29 \( 1 - 8.66iT - 29T^{2} \)
31 \( 1 - 3.16iT - 31T^{2} \)
37 \( 1 - 3.79iT - 37T^{2} \)
41 \( 1 - 5.95T + 41T^{2} \)
43 \( 1 + 7.87T + 43T^{2} \)
47 \( 1 - 6.85T + 47T^{2} \)
53 \( 1 + 12.4iT - 53T^{2} \)
59 \( 1 + 1.69iT - 59T^{2} \)
61 \( 1 - 7.32T + 61T^{2} \)
67 \( 1 - 1.64iT - 67T^{2} \)
71 \( 1 + 4.95T + 71T^{2} \)
73 \( 1 - 13.6iT - 73T^{2} \)
79 \( 1 - 2.01iT - 79T^{2} \)
83 \( 1 + 3.14iT - 83T^{2} \)
89 \( 1 - 4.88iT - 89T^{2} \)
97 \( 1 + 15.6T + 97T^{2} \)
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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

−10.29838890426476362916939209234, −8.624725795998972067288997884057, −8.063564799999792701506525777367, −6.75477955800675303846974382197, −5.83847062287216726968592958973, −5.40237337433377213518467349283, −4.88118622553327083611559720161, −3.76826074085582627705044061459, −2.59356694300469214036111274084, −1.45084874211135472665860840381, 1.95140578687909373868967349954, 2.53879691293190107937021532309, 4.17595454138091361713150762196, 4.57950886232579793371417560040, 5.49947956749934430964818293642, 6.27765365921367574807744745707, 7.03273092155500633772442477003, 7.61428609910566119006664696403, 9.259258686681256106812438324946, 10.26143964344972427409236964185

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