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Properties

Label 2-1155-385.384-c1-0-10
Degree $2$
Conductor $1155$
Sign $-0.0105 - 0.999i$
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  − 1.63·2-s − 3-s + 0.674·4-s + (1.84 − 1.25i)5-s + 1.63·6-s + (1.79 − 1.94i)7-s + 2.16·8-s + 9-s + (−3.02 + 2.05i)10-s + (−3.27 + 0.520i)11-s − 0.674·12-s + 4.17i·13-s + (−2.93 + 3.18i)14-s + (−1.84 + 1.25i)15-s − 4.89·16-s + 7.58i·17-s + ⋯
L(s)  = 1  − 1.15·2-s − 0.577·3-s + 0.337·4-s + (0.826 − 0.562i)5-s + 0.667·6-s + (0.677 − 0.735i)7-s + 0.766·8-s + 0.333·9-s + (−0.956 + 0.650i)10-s + (−0.987 + 0.156i)11-s − 0.194·12-s + 1.15i·13-s + (−0.783 + 0.850i)14-s + (−0.477 + 0.324i)15-s − 1.22·16-s + 1.83i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0105 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0105 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.0105 - 0.999i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ -0.0105 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4671417354\)
\(L(\frac12)\) \(\approx\) \(0.4671417354\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + (-1.84 + 1.25i)T \)
7 \( 1 + (-1.79 + 1.94i)T \)
11 \( 1 + (3.27 - 0.520i)T \)
good2 \( 1 + 1.63T + 2T^{2} \)
13 \( 1 - 4.17iT - 13T^{2} \)
17 \( 1 - 7.58iT - 17T^{2} \)
19 \( 1 + 5.73T + 19T^{2} \)
23 \( 1 + 2.17iT - 23T^{2} \)
29 \( 1 - 5.39iT - 29T^{2} \)
31 \( 1 + 0.864iT - 31T^{2} \)
37 \( 1 - 2.91iT - 37T^{2} \)
41 \( 1 - 3.36T + 41T^{2} \)
43 \( 1 - 5.31T + 43T^{2} \)
47 \( 1 + 10.7T + 47T^{2} \)
53 \( 1 - 11.1iT - 53T^{2} \)
59 \( 1 - 10.8iT - 59T^{2} \)
61 \( 1 + 15.3T + 61T^{2} \)
67 \( 1 + 10.6iT - 67T^{2} \)
71 \( 1 + 1.55T + 71T^{2} \)
73 \( 1 - 2.25iT - 73T^{2} \)
79 \( 1 - 8.24iT - 79T^{2} \)
83 \( 1 - 5.54iT - 83T^{2} \)
89 \( 1 - 1.62iT - 89T^{2} \)
97 \( 1 + 8.80T + 97T^{2} \)
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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

−10.10447763944145492292070509391, −9.131501001021233420591961870135, −8.466019687869913061004276538222, −7.78958382573897928533898891591, −6.76817241175840545818193202699, −5.94397446098308930645531699721, −4.70028201605878567968601593739, −4.29926915240925919845728043792, −2.02869311236577523892820914665, −1.31393377189021802920262862535, 0.34046534478096257522791122996, 1.93385378860914214363400166769, 2.84443653248125293163327697924, 4.75092444400777371095196006634, 5.37811526949934250168536222496, 6.25659438398621694027515123278, 7.39437392936706412284054420064, 7.960922759413967672272716598947, 8.871526247185035301118738659715, 9.683342196434555030732902441427

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