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Properties

Label 2-1155-5.4-c1-0-14
Degree $2$
Conductor $1155$
Sign $0.0894 - 0.995i$
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  − 0.792i·2-s + i·3-s + 1.37·4-s + (−0.199 + 2.22i)5-s + 0.792·6-s + i·7-s − 2.67i·8-s − 9-s + (1.76 + 0.158i)10-s + 11-s + 1.37i·12-s + 6.97i·13-s + 0.792·14-s + (−2.22 − 0.199i)15-s + 0.629·16-s + 2.01i·17-s + ⋯
L(s)  = 1  − 0.560i·2-s + 0.577i·3-s + 0.686·4-s + (−0.0894 + 0.995i)5-s + 0.323·6-s + 0.377i·7-s − 0.944i·8-s − 0.333·9-s + (0.557 + 0.0500i)10-s + 0.301·11-s + 0.396i·12-s + 1.93i·13-s + 0.211·14-s + (−0.575 − 0.0516i)15-s + 0.157·16-s + 0.487i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0894 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0894 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.0894 - 0.995i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.0894 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.690975452\)
\(L(\frac12)\) \(\approx\) \(1.690975452\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.199 - 2.22i)T \)
7 \( 1 - iT \)
11 \( 1 - T \)
good2 \( 1 + 0.792iT - 2T^{2} \)
13 \( 1 - 6.97iT - 13T^{2} \)
17 \( 1 - 2.01iT - 17T^{2} \)
19 \( 1 + 6.64T + 19T^{2} \)
23 \( 1 + 2.05iT - 23T^{2} \)
29 \( 1 - 6.58T + 29T^{2} \)
31 \( 1 + 6.71T + 31T^{2} \)
37 \( 1 - 4.53iT - 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + 5.38iT - 43T^{2} \)
47 \( 1 - 1.71iT - 47T^{2} \)
53 \( 1 - 5.42iT - 53T^{2} \)
59 \( 1 + 7.79T + 59T^{2} \)
61 \( 1 + 2.21T + 61T^{2} \)
67 \( 1 - 2.45iT - 67T^{2} \)
71 \( 1 - 6.95T + 71T^{2} \)
73 \( 1 - 9.73iT - 73T^{2} \)
79 \( 1 - 6.67T + 79T^{2} \)
83 \( 1 - 4.69iT - 83T^{2} \)
89 \( 1 + 6.34T + 89T^{2} \)
97 \( 1 + 15.0iT - 97T^{2} \)
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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

−10.18920359418584522671113930471, −9.346247028691401402495224652358, −8.569089431702443704346735887777, −7.33592248377808331889445133044, −6.50519204205394510603680682200, −6.11517477482945090609688818862, −4.42615781555748547149766479463, −3.79590776120231361403881060507, −2.63224084995885194053628264248, −1.88369128802266645130895393696, 0.69538665690409179076981111677, 2.03293793215436346862906378618, 3.24933777639765887253642274734, 4.62406351863490785734023989737, 5.59552676786886394957727598933, 6.19722502076195779189920550497, 7.24004944390691596656532210512, 7.913268325546935869950252121930, 8.422385020954384710243394197421, 9.424620311283214884720680398758

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