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Properties

Label 2-1155-5.4-c1-0-42
Degree $2$
Conductor $1155$
Sign $0.165 + 0.986i$
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  − 0.332i·2-s + i·3-s + 1.88·4-s + (−0.370 − 2.20i)5-s + 0.332·6-s − i·7-s − 1.29i·8-s − 9-s + (−0.733 + 0.123i)10-s − 11-s + 1.88i·12-s − 5.50i·13-s − 0.332·14-s + (2.20 − 0.370i)15-s + 3.34·16-s + 6.85i·17-s + ⋯
L(s)  = 1  − 0.235i·2-s + 0.577i·3-s + 0.944·4-s + (−0.165 − 0.986i)5-s + 0.135·6-s − 0.377i·7-s − 0.457i·8-s − 0.333·9-s + (−0.231 + 0.0389i)10-s − 0.301·11-s + 0.545i·12-s − 1.52i·13-s − 0.0888·14-s + (0.569 − 0.0957i)15-s + 0.837·16-s + 1.66i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.165 + 0.986i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (694, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.165 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.803327992\)
\(L(\frac12)\) \(\approx\) \(1.803327992\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 + (0.370 + 2.20i)T \)
7 \( 1 + iT \)
11 \( 1 + T \)
good2 \( 1 + 0.332iT - 2T^{2} \)
13 \( 1 + 5.50iT - 13T^{2} \)
17 \( 1 - 6.85iT - 17T^{2} \)
19 \( 1 - 3.55T + 19T^{2} \)
23 \( 1 + 8.38iT - 23T^{2} \)
29 \( 1 + 8.24T + 29T^{2} \)
31 \( 1 - 0.581T + 31T^{2} \)
37 \( 1 + 9.80iT - 37T^{2} \)
41 \( 1 - 6.19T + 41T^{2} \)
43 \( 1 + 3.22iT - 43T^{2} \)
47 \( 1 + 6.73iT - 47T^{2} \)
53 \( 1 - 14.0iT - 53T^{2} \)
59 \( 1 + 2.05T + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 - 1.52iT - 67T^{2} \)
71 \( 1 + 2.12T + 71T^{2} \)
73 \( 1 - 1.03iT - 73T^{2} \)
79 \( 1 - 1.77T + 79T^{2} \)
83 \( 1 - 4.43iT - 83T^{2} \)
89 \( 1 + 2.23T + 89T^{2} \)
97 \( 1 + 11.2iT - 97T^{2} \)
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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

−9.806942764196632804321473996376, −8.754809278920652467124329874669, −7.996132144997069704195314088359, −7.33066765194179014867248517803, −5.93862664301412358058970931141, −5.49488174721335086641944957218, −4.21740052858116613309575538072, −3.45352131781296118392414957520, −2.20780897863573515660634990888, −0.76210947335356037926452585853, 1.69951702566412513874909434510, 2.64286515473304398819710000189, 3.48630796354956287045724748930, 5.10580722530882707477118630231, 6.00343144358689322194194881662, 6.84884814958717001768357996851, 7.33016341136069753686918759254, 7.926540508268924429759512015235, 9.295129131252319503714619823913, 9.843446574582917213464501496980

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