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Properties

Label 2-1452-33.32-c1-0-11
Degree $2$
Conductor $1452$
Sign $-0.548 - 0.836i$
Analytic cond. $11.5942$
Root an. cond. $3.40503$
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  + (1.24 + 1.20i)3-s + 3.05i·5-s − 3.65i·7-s + (0.0930 + 2.99i)9-s + 1.08i·13-s + (−3.68 + 3.80i)15-s − 1.23·17-s + 5.17i·19-s + (4.40 − 4.53i)21-s + 6.43i·23-s − 4.35·25-s + (−3.49 + 3.84i)27-s − 7.42·29-s + 3.91·31-s + 11.1·35-s + ⋯
L(s)  = 1  + (0.717 + 0.696i)3-s + 1.36i·5-s − 1.37i·7-s + (0.0310 + 0.999i)9-s + 0.299i·13-s + (−0.952 + 0.982i)15-s − 0.298·17-s + 1.18i·19-s + (0.960 − 0.990i)21-s + 1.34i·23-s − 0.871·25-s + (−0.673 + 0.739i)27-s − 1.37·29-s + 0.703·31-s + 1.88·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 - 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.548 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $-0.548 - 0.836i$
Analytic conductor: \(11.5942\)
Root analytic conductor: \(3.40503\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1452} (725, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1452,\ (\ :1/2),\ -0.548 - 0.836i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.851548458\)
\(L(\frac12)\) \(\approx\) \(1.851548458\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.24 - 1.20i)T \)
11 \( 1 \)
good5 \( 1 - 3.05iT - 5T^{2} \)
7 \( 1 + 3.65iT - 7T^{2} \)
13 \( 1 - 1.08iT - 13T^{2} \)
17 \( 1 + 1.23T + 17T^{2} \)
19 \( 1 - 5.17iT - 19T^{2} \)
23 \( 1 - 6.43iT - 23T^{2} \)
29 \( 1 + 7.42T + 29T^{2} \)
31 \( 1 - 3.91T + 31T^{2} \)
37 \( 1 - 0.211T + 37T^{2} \)
41 \( 1 - 6.43T + 41T^{2} \)
43 \( 1 - 11.6iT - 43T^{2} \)
47 \( 1 + 3.18iT - 47T^{2} \)
53 \( 1 + 8.20iT - 53T^{2} \)
59 \( 1 - 6.23iT - 59T^{2} \)
61 \( 1 - 6.10iT - 61T^{2} \)
67 \( 1 + 3.24T + 67T^{2} \)
71 \( 1 + 0.332iT - 71T^{2} \)
73 \( 1 + 13.6iT - 73T^{2} \)
79 \( 1 - 1.32iT - 79T^{2} \)
83 \( 1 - 7.81T + 83T^{2} \)
89 \( 1 + 10.3iT - 89T^{2} \)
97 \( 1 - 6.35T + 97T^{2} \)
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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

−9.890247626063246371814627149773, −9.232817403418016097653624581823, −7.910765089545736880568472707118, −7.54773407071326859713157129099, −6.73439371988802709812612907477, −5.71647320921139344479850785489, −4.39376825489295683966007493128, −3.72077005152524726333739869575, −3.03752155809882850208950075895, −1.76071897253133705624458145859, 0.66388163876740529341851027068, 2.03876991746229687907920456807, 2.78268301234032935699765647586, 4.17026530991707768403839332065, 5.12884875331101106015384906253, 5.90730629568820286460978426245, 6.84589068639110424535294018734, 7.905152422920815705758507553038, 8.597437834968493660632064206898, 9.030568676334477303778834459697

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