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Properties

Label 2-1452-11.5-c1-0-10
Degree $2$
Conductor $1452$
Sign $0.780 - 0.625i$
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  + (0.809 + 0.587i)3-s + (−0.309 + 0.951i)5-s + (1.61 − 1.17i)7-s + (0.309 + 0.951i)9-s + (0.927 + 2.85i)13-s + (−0.809 + 0.587i)15-s + (2.16 − 6.65i)17-s + (3.23 + 2.35i)19-s + 2·21-s − 2·23-s + (3.23 + 2.35i)25-s + (−0.309 + 0.951i)27-s + (2.42 − 1.76i)29-s + (0.618 + 1.90i)35-s + (−7.28 + 5.29i)37-s + ⋯
L(s)  = 1  + (0.467 + 0.339i)3-s + (−0.138 + 0.425i)5-s + (0.611 − 0.444i)7-s + (0.103 + 0.317i)9-s + (0.257 + 0.791i)13-s + (−0.208 + 0.151i)15-s + (0.524 − 1.61i)17-s + (0.742 + 0.539i)19-s + 0.436·21-s − 0.417·23-s + (0.647 + 0.470i)25-s + (−0.0594 + 0.183i)27-s + (0.450 − 0.327i)29-s + (0.104 + 0.321i)35-s + (−1.19 + 0.869i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 - 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $0.780 - 0.625i$
Analytic conductor: \(11.5942\)
Root analytic conductor: \(3.40503\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1452} (1237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1452,\ (\ :1/2),\ 0.780 - 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.177797445\)
\(L(\frac12)\) \(\approx\) \(2.177797445\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 \)
good5 \( 1 + (0.309 - 0.951i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.61 + 1.17i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.927 - 2.85i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.16 + 6.65i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.23 - 2.35i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 + (-2.42 + 1.76i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (7.28 - 5.29i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-7.28 - 5.29i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-9.70 - 7.05i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (4.01 + 12.3i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-4.85 + 3.52i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.618 + 1.90i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 14T + 67T^{2} \)
71 \( 1 + (1.85 - 5.70i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.85 + 3.52i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.94 - 15.2i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.47 + 7.60i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - T + 89T^{2} \)
97 \( 1 + (-2.16 - 6.65i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.600470828033869045477987226244, −8.892763958234161492525351651567, −7.87068781070234371499763706522, −7.37942857363695551121911500395, −6.48084814825939068290475413707, −5.26659738041004497230250985587, −4.51616793721705409782307932799, −3.53623961696262903983289177339, −2.64096913159027827524216281389, −1.25610116474299648280681821902, 1.01271914629319192854901496950, 2.16730898191930565994435109693, 3.31377370831796005736407263412, 4.28439577093229492829562402642, 5.38611627620177608179359781950, 6.01704920473596091899393198831, 7.23553402489200625968281518208, 7.921409010209840653025838367108, 8.650137470927553058153429460796, 9.091967734966419414752926284836

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