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Properties

Label 2-1155-33.32-c1-0-53
Degree $2$
Conductor $1155$
Sign $0.955 + 0.296i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.955 + 0.296i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.955 + 0.296i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.145674342\)
\(L(\frac12)\) \(\approx\) \(1.145674342\)
\(L(\frac{3}{2})\) 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

−9.697794297056150172387400842933, −8.907037966711842340000706103283, −8.246114624112202311617644669030, −7.69631747203728205802702486728, −6.92823358148386921902783052219, −5.85820074179400204175900936157, −4.26932799931227212045811048194, −3.23982904050223774992983722676, −1.92799472958681413806196994930, −1.02618627916672889099187323600, 1.12606498963434621840932104730, 2.25511632364159099375098064924, 3.07725190676827925186936670634, 4.42604579724940515128931002651, 6.26214550850089093628300029666, 6.87089527146202445857776769781, 7.56994687704450209285671785354, 8.400571741023443878754633586621, 9.233364923767503171432393130062, 9.336644174341597425113742874378

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