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Properties

Label 2-1155-33.32-c1-0-89
Degree $2$
Conductor $1155$
Sign $0.998 + 0.0602i$
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  + 2.50·2-s + (1.71 − 0.207i)3-s + 4.29·4-s + i·5-s + (4.31 − 0.520i)6-s − i·7-s + 5.75·8-s + (2.91 − 0.713i)9-s + 2.50i·10-s + (−3.26 − 0.595i)11-s + (7.38 − 0.890i)12-s − 2.68i·13-s − 2.50i·14-s + (0.207 + 1.71i)15-s + 5.84·16-s − 1.44·17-s + ⋯
L(s)  = 1  + 1.77·2-s + (0.992 − 0.119i)3-s + 2.14·4-s + 0.447i·5-s + (1.76 − 0.212i)6-s − 0.377i·7-s + 2.03·8-s + (0.971 − 0.237i)9-s + 0.793i·10-s + (−0.983 − 0.179i)11-s + (2.13 − 0.257i)12-s − 0.744i·13-s − 0.670i·14-s + (0.0535 + 0.443i)15-s + 1.46·16-s − 0.351·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.998 + 0.0602i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.998 + 0.0602i)\)

Particular Values

\(L(1)\) \(\approx\) \(6.116246512\)
\(L(\frac12)\) \(\approx\) \(6.116246512\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.71 + 0.207i)T \)
5 \( 1 - iT \)
7 \( 1 + iT \)
11 \( 1 + (3.26 + 0.595i)T \)
good2 \( 1 - 2.50T + 2T^{2} \)
13 \( 1 + 2.68iT - 13T^{2} \)
17 \( 1 + 1.44T + 17T^{2} \)
19 \( 1 - 7.09iT - 19T^{2} \)
23 \( 1 - 0.250iT - 23T^{2} \)
29 \( 1 + 4.88T + 29T^{2} \)
31 \( 1 + 4.26T + 31T^{2} \)
37 \( 1 - 7.50T + 37T^{2} \)
41 \( 1 + 1.94T + 41T^{2} \)
43 \( 1 + 7.29iT - 43T^{2} \)
47 \( 1 + 1.14iT - 47T^{2} \)
53 \( 1 - 2.58iT - 53T^{2} \)
59 \( 1 - 11.7iT - 59T^{2} \)
61 \( 1 - 13.2iT - 61T^{2} \)
67 \( 1 + 5.66T + 67T^{2} \)
71 \( 1 + 4.04iT - 71T^{2} \)
73 \( 1 + 0.248iT - 73T^{2} \)
79 \( 1 + 14.3iT - 79T^{2} \)
83 \( 1 - 3.49T + 83T^{2} \)
89 \( 1 + 11.0iT - 89T^{2} \)
97 \( 1 + 2.11T + 97T^{2} \)
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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

−10.20851869007616692035513607298, −8.827304857341281974431518185468, −7.58654566565142737766157015849, −7.45911767275704626853253536496, −6.16795719304664775236488912837, −5.50745883550119933310817722787, −4.34538259031742211850613305651, −3.58381529144474164363297392843, −2.87389141232327783050950234879, −1.90002132692161126038810855298, 2.01030765558351209814738924225, 2.69675661646298638358104477020, 3.70807336244832689987661054418, 4.67840375761071163251119889459, 5.10253058888537497761399169691, 6.31614211906818524229766483885, 7.15310643387999131336931231934, 7.966339120283719792829567593043, 9.021775241545991477480114795503, 9.699871270723552373482007850916

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