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Properties

Label 2-1452-11.5-c1-0-13
Degree $2$
Conductor $1452$
Sign $0.836 + 0.548i$
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 + (0.881 − 2.71i)5-s + (3.42 − 2.48i)7-s + (0.309 + 0.951i)9-s + (0.545 + 1.67i)13-s + (2.30 − 1.67i)15-s + (−1.42 + 4.39i)17-s + (4.92 + 3.57i)19-s + 4.23·21-s + 4.23·23-s + (−2.54 − 1.84i)25-s + (−0.309 + 0.951i)27-s + (−3.61 + 2.62i)29-s + (−2.66 − 8.19i)31-s + (−3.73 − 11.4i)35-s + ⋯
L(s)  = 1  + (0.467 + 0.339i)3-s + (0.394 − 1.21i)5-s + (1.29 − 0.941i)7-s + (0.103 + 0.317i)9-s + (0.151 + 0.465i)13-s + (0.596 − 0.433i)15-s + (−0.346 + 1.06i)17-s + (1.13 + 0.821i)19-s + 0.924·21-s + 0.883·23-s + (−0.509 − 0.369i)25-s + (−0.0594 + 0.183i)27-s + (−0.671 + 0.488i)29-s + (−0.478 − 1.47i)31-s + (−0.631 − 1.94i)35-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $0.836 + 0.548i$
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: \((2,\ 1452,\ (\ :1/2),\ 0.836 + 0.548i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.562142970\)
\(L(\frac12)\) \(\approx\) \(2.562142970\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.881 + 2.71i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-3.42 + 2.48i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.545 - 1.67i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.42 - 4.39i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-4.92 - 3.57i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 4.23T + 23T^{2} \)
29 \( 1 + (3.61 - 2.62i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.66 + 8.19i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.66 + 4.84i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-0.427 - 0.310i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 0.527T + 43T^{2} \)
47 \( 1 + (1.11 + 0.812i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (4.19 + 12.8i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.16 - 5.20i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.118 + 0.363i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 6.85T + 67T^{2} \)
71 \( 1 + (1.11 - 3.44i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (1 - 0.726i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.01 + 9.28i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.01 - 6.20i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - T + 89T^{2} \)
97 \( 1 + (-1.88 - 5.79i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.329578180748982655595233762199, −8.693892409751207178562866351630, −7.912886401503167878770301356808, −7.35905432226750391417861408338, −5.94574767110797020533495865719, −5.09250821540177995950010502438, −4.39268154961595935675342281295, −3.65920475798492203702218745827, −1.92530672836687330750981561592, −1.17789653036762856254497972788, 1.44361712997005719225404500877, 2.67852532004189957296935906325, 3.03184623066341338528922963449, 4.70709157147341572135878207183, 5.44178085925122388693932892792, 6.43577639545123256152269579587, 7.30681263741630283676886798192, 7.82989076310910456400611431815, 8.906866022007228898780621377534, 9.367041829270972755930063608037

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