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Properties

Label 2-1155-5.4-c1-0-10
Degree $2$
Conductor $1155$
Sign $-0.894 - 0.447i$
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  + i·2-s + i·3-s + 4-s + (−2 − i)5-s − 6-s − i·7-s + 3i·8-s − 9-s + (1 − 2i)10-s + 11-s + i·12-s + 14-s + (1 − 2i)15-s − 16-s + 6i·17-s − i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s + 0.5·4-s + (−0.894 − 0.447i)5-s − 0.408·6-s − 0.377i·7-s + 1.06i·8-s − 0.333·9-s + (0.316 − 0.632i)10-s + 0.301·11-s + 0.288i·12-s + 0.267·14-s + (0.258 − 0.516i)15-s − 0.250·16-s + 1.45i·17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.894 - 0.447i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.299842771\)
\(L(\frac12)\) \(\approx\) \(1.299842771\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + i)T \)
7 \( 1 + iT \)
11 \( 1 - T \)
good2 \( 1 - iT - 2T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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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

−10.19919550171989171630485507957, −9.110530735907319921264441673417, −8.344588168152005753671927811958, −7.69727475863075152313665627912, −6.91541796721221392793059397525, −5.93670099945082156465073246983, −5.11299480821682757260827718333, −4.06802003649711077096787991795, −3.30484578069762892950136960942, −1.61859760466338375287059253568, 0.55170394821371558821969932425, 2.10154163342333559636944188902, 2.94648901688628314398379962575, 3.82994388585028736148059895500, 5.07551724314611903166488882426, 6.38533601026859193980550670572, 6.97251080753238389549594678385, 7.65130469186450579823583422549, 8.626337533542103104125760790691, 9.535734241313627084831169393116

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