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Properties

Label 2-1155-1155.1154-c0-0-9
Degree $2$
Conductor $1155$
Sign $i$
Analytic cond. $0.576420$
Root an. cond. $0.759223$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s + 4-s − i·5-s + i·7-s − 9-s − 11-s − i·12-s − 2i·13-s − 15-s + 16-s − i·20-s + 21-s − 25-s + i·27-s + i·28-s + 2·29-s + ⋯
L(s)  = 1  − i·3-s + 4-s − i·5-s + i·7-s − 9-s − 11-s − i·12-s − 2i·13-s − 15-s + 16-s − i·20-s + 21-s − 25-s + i·27-s + i·28-s + 2·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $i$
Analytic conductor: \(0.576420\)
Root analytic conductor: \(0.759223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (1154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.204523816\)
\(L(\frac12)\) \(\approx\) \(1.204523816\)
\(L(1)\) 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 \( 1 + iT \)
7 \( 1 - iT \)
11 \( 1 + T \)
good2 \( 1 - T^{2} \)
13 \( 1 + 2iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - 2T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 2iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 2iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.850256852350633614505772997613, −8.550525623795299051862865294229, −8.125603444344571218299691627371, −7.52695876646109655970032872526, −6.29497973143128202937679239261, −5.66669141481264212390760281664, −5.04378466490444627673788989768, −3.01317513250868531146554874928, −2.48821146981452051842978688500, −1.11280477851658617413417186044, 2.11137506925356803682352307925, 3.08710890079369849769592324563, 3.99154359024583216622674382151, 4.94468631215194816503747846857, 6.24306152334201981136965677937, 6.79740997810864848951290662490, 7.57998966343385209913127258383, 8.550831659664230657872268834606, 9.759502643246273420214755907651, 10.33095421162463641514248571015

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