Properties

Label 2-1155-33.32-c1-0-55
Degree $2$
Conductor $1155$
Sign $0.646 + 0.762i$
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  + 0.805·2-s + (−0.972 + 1.43i)3-s − 1.35·4-s + i·5-s + (−0.783 + 1.15i)6-s + i·7-s − 2.69·8-s + (−1.10 − 2.78i)9-s + 0.805i·10-s + (0.889 − 3.19i)11-s + (1.31 − 1.93i)12-s − 0.669i·13-s + 0.805i·14-s + (−1.43 − 0.972i)15-s + 0.530·16-s − 5.06·17-s + ⋯
L(s)  = 1  + 0.569·2-s + (−0.561 + 0.827i)3-s − 0.675·4-s + 0.447i·5-s + (−0.319 + 0.471i)6-s + 0.377i·7-s − 0.954·8-s + (−0.369 − 0.929i)9-s + 0.254i·10-s + (0.268 − 0.963i)11-s + (0.379 − 0.559i)12-s − 0.185i·13-s + 0.215i·14-s + (−0.370 − 0.251i)15-s + 0.132·16-s − 1.22·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.646 + 0.762i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.646 + 0.762i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8565442475\)
\(L(\frac12)\) \(\approx\) \(0.8565442475\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.972 - 1.43i)T \)
5 \( 1 - iT \)
7 \( 1 - iT \)
11 \( 1 + (-0.889 + 3.19i)T \)
good2 \( 1 - 0.805T + 2T^{2} \)
13 \( 1 + 0.669iT - 13T^{2} \)
17 \( 1 + 5.06T + 17T^{2} \)
19 \( 1 + 8.50iT - 19T^{2} \)
23 \( 1 - 6.77iT - 23T^{2} \)
29 \( 1 - 2.44T + 29T^{2} \)
31 \( 1 - 5.73T + 31T^{2} \)
37 \( 1 + 4.26T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 - 1.79iT - 43T^{2} \)
47 \( 1 + 7.04iT - 47T^{2} \)
53 \( 1 + 9.79iT - 53T^{2} \)
59 \( 1 + 11.6iT - 59T^{2} \)
61 \( 1 + 7.58iT - 61T^{2} \)
67 \( 1 + 9.11T + 67T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 + 12.3iT - 73T^{2} \)
79 \( 1 + 11.9iT - 79T^{2} \)
83 \( 1 + 6.36T + 83T^{2} \)
89 \( 1 - 8.93iT - 89T^{2} \)
97 \( 1 + 9.76T + 97T^{2} \)
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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

−9.531670035317055599764143012549, −9.075930672025273354069053370951, −8.323171804538863374627104343318, −6.80970645716366094474431275601, −6.12333744971131964307926493860, −5.25024305538391880817691101113, −4.58463386980908445109864254773, −3.60139710980270979321919171013, −2.79591802750606216814509141565, −0.38202455626171568084084902567, 1.20294302346511101863426640694, 2.55249018821733711805943363872, 4.29327112415705442563286993725, 4.49472532951460344383465956196, 5.75204113544588920414142705127, 6.35940169596177854625509607583, 7.33813340664007630387696413838, 8.255760865245816445411898969292, 8.945086137564775047370092062685, 10.00292996632004550864096786632

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