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Properties

Label 2-1155-385.384-c1-0-32
Degree $2$
Conductor $1155$
Sign $0.705 + 0.708i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.705 + 0.708i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.705 + 0.708i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4354808271\)
\(L(\frac12)\) \(\approx\) \(0.4354808271\)
\(L(\frac{3}{2})\) 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

−9.608605367075894759512186084838, −8.851105278570876681218621205076, −8.318696301281805824361383727520, −7.48370887378276295372805936475, −6.48881020414502733209082825873, −5.33312529343631087571620114075, −4.86973960599680353022021297404, −3.29096907913952870174144737231, −1.76983198972415608303462224784, −0.51710440551429462699111758141, 0.74260269692214077784131644066, 2.31711750933535736128827131801, 3.65458301829852809084386687228, 4.70770957422246697430307569245, 5.96311609423406401509121343605, 6.93800977880139739931451070065, 7.57702013643231984549089423459, 7.956533746008264073420900862204, 9.385292115757463360239240789127, 10.02215983066646565420020194246

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