Properties

Label 2-1155-1155.734-c0-0-2
Degree $2$
Conductor $1155$
Sign $-0.766 + 0.642i$
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  + (−0.587 − 0.809i)3-s + (0.309 − 0.951i)4-s + (0.587 − 0.809i)5-s + (−0.951 − 0.309i)7-s + (−0.309 + 0.951i)9-s + (0.809 − 0.587i)11-s + (−0.951 + 0.309i)12-s + (−0.363 − 0.5i)13-s − 15-s + (−0.809 − 0.587i)16-s + (0.951 + 0.690i)17-s + (−0.587 − 0.809i)20-s + (0.309 + 0.951i)21-s + (−0.309 − 0.951i)25-s + (0.951 − 0.309i)27-s + (−0.587 + 0.809i)28-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)3-s + (0.309 − 0.951i)4-s + (0.587 − 0.809i)5-s + (−0.951 − 0.309i)7-s + (−0.309 + 0.951i)9-s + (0.809 − 0.587i)11-s + (−0.951 + 0.309i)12-s + (−0.363 − 0.5i)13-s − 15-s + (−0.809 − 0.587i)16-s + (0.951 + 0.690i)17-s + (−0.587 − 0.809i)20-s + (0.309 + 0.951i)21-s + (−0.309 − 0.951i)25-s + (0.951 − 0.309i)27-s + (−0.587 + 0.809i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(0.576420\)
Root analytic conductor: \(0.759223\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (734, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :0),\ -0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8965683037\)
\(L(\frac12)\) \(\approx\) \(0.8965683037\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.587 + 0.809i)T \)
5 \( 1 + (-0.587 + 0.809i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (-0.809 + 0.587i)T \)
good2 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.309 + 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \)
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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

−9.761085017462901277048154969510, −9.005506030792556949439757150119, −7.956215127767609554798944086352, −6.88961153795617002771612454202, −6.22569441396110613870537846069, −5.65558200291929615643508290232, −4.86859738626285699447213127543, −3.30205462792196647919216545507, −1.81725985449129133163190639007, −0.883182437990402172295430772671, 2.31535240084069125657915237706, 3.31784775946854281687624922032, 4.01572193588355142139484238141, 5.25918221459381387521170654537, 6.35064742725919505558790733772, 6.73528243182721311513669069515, 7.67650558217900215721170146888, 9.008472707855024382793025604575, 9.687371003254752550258689224818, 10.03819528264774275741600589637

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