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Properties

Label 2-1155-33.32-c1-0-71
Degree $2$
Conductor $1155$
Sign $0.837 + 0.546i$
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  + 2.72·2-s + (−1.44 − 0.948i)3-s + 5.40·4-s − i·5-s + (−3.94 − 2.58i)6-s + i·7-s + 9.27·8-s + (1.20 + 2.74i)9-s − 2.72i·10-s + (3.31 − 0.00239i)11-s + (−7.83 − 5.12i)12-s + 0.172i·13-s + 2.72i·14-s + (−0.948 + 1.44i)15-s + 14.4·16-s − 0.805·17-s + ⋯
L(s)  = 1  + 1.92·2-s + (−0.836 − 0.547i)3-s + 2.70·4-s − 0.447i·5-s + (−1.61 − 1.05i)6-s + 0.377i·7-s + 3.27·8-s + (0.400 + 0.916i)9-s − 0.860i·10-s + (0.999 − 0.000720i)11-s + (−2.26 − 1.47i)12-s + 0.0477i·13-s + 0.727i·14-s + (−0.244 + 0.374i)15-s + 3.60·16-s − 0.195·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.837 + 0.546i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.837 + 0.546i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.486495648\)
\(L(\frac12)\) \(\approx\) \(4.486495648\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.44 + 0.948i)T \)
5 \( 1 + iT \)
7 \( 1 - iT \)
11 \( 1 + (-3.31 + 0.00239i)T \)
good2 \( 1 - 2.72T + 2T^{2} \)
13 \( 1 - 0.172iT - 13T^{2} \)
17 \( 1 + 0.805T + 17T^{2} \)
19 \( 1 + 2.99iT - 19T^{2} \)
23 \( 1 - 0.277iT - 23T^{2} \)
29 \( 1 + 7.75T + 29T^{2} \)
31 \( 1 - 2.70T + 31T^{2} \)
37 \( 1 + 1.15T + 37T^{2} \)
41 \( 1 - 2.26T + 41T^{2} \)
43 \( 1 + 9.92iT - 43T^{2} \)
47 \( 1 - 11.2iT - 47T^{2} \)
53 \( 1 + 10.5iT - 53T^{2} \)
59 \( 1 - 5.52iT - 59T^{2} \)
61 \( 1 - 4.15iT - 61T^{2} \)
67 \( 1 + 8.68T + 67T^{2} \)
71 \( 1 - 14.8iT - 71T^{2} \)
73 \( 1 - 6.17iT - 73T^{2} \)
79 \( 1 - 3.40iT - 79T^{2} \)
83 \( 1 + 18.1T + 83T^{2} \)
89 \( 1 + 14.7iT - 89T^{2} \)
97 \( 1 - 15.5T + 97T^{2} \)
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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

−10.07333103629596362833296591222, −8.789692415318506116437221811126, −7.49623964022794943665757656868, −6.88190171370984079081138655487, −6.05438947835642132709462119017, −5.47567004739467595199746289558, −4.62583777697747100039274917516, −3.86298305864858725435189437994, −2.49815859405687862740465394361, −1.44228751396451795895876352637, 1.64566781931176390776877190866, 3.19097469638711316063956870879, 3.93205035610717472206658612523, 4.57281053847736541664257773224, 5.57922695354276326963063836168, 6.24814010042647475643069708109, 6.84876340927403146176512933442, 7.71845951777887798974498601768, 9.365003791922970901628852296826, 10.34385372169163335811762073379

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