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Properties

Label 2-1512-63.4-c1-0-16
Degree $2$
Conductor $1512$
Sign $0.998 + 0.0535i$
Analytic cond. $12.0733$
Root an. cond. $3.47467$
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  + 3.79·5-s + (2.59 + 0.525i)7-s − 4.51·11-s + (0.588 − 1.01i)13-s + (2.95 − 5.12i)17-s + (2.55 + 4.42i)19-s + 4.18·23-s + 9.43·25-s + (−2.11 − 3.65i)29-s + (3.12 + 5.40i)31-s + (9.85 + 1.99i)35-s + (−3.87 − 6.70i)37-s + (−0.754 + 1.30i)41-s + (−5.01 − 8.68i)43-s + (−1.11 + 1.93i)47-s + ⋯
L(s)  = 1  + 1.69·5-s + (0.980 + 0.198i)7-s − 1.36·11-s + (0.163 − 0.282i)13-s + (0.717 − 1.24i)17-s + (0.586 + 1.01i)19-s + 0.871·23-s + 1.88·25-s + (−0.392 − 0.679i)29-s + (0.560 + 0.971i)31-s + (1.66 + 0.337i)35-s + (−0.636 − 1.10i)37-s + (−0.117 + 0.204i)41-s + (−0.765 − 1.32i)43-s + (−0.163 + 0.282i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.998 + 0.0535i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.998 + 0.0535i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.547296854\)
\(L(\frac12)\) \(\approx\) \(2.547296854\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-2.59 - 0.525i)T \)
good5 \( 1 - 3.79T + 5T^{2} \)
11 \( 1 + 4.51T + 11T^{2} \)
13 \( 1 + (-0.588 + 1.01i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.95 + 5.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.55 - 4.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.18T + 23T^{2} \)
29 \( 1 + (2.11 + 3.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.12 - 5.40i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.87 + 6.70i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.754 - 1.30i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.01 + 8.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.11 - 1.93i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.49 - 11.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.19 - 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.729 - 1.26i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.813 + 1.40i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.48T + 71T^{2} \)
73 \( 1 + (-3.72 + 6.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.920 + 1.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.307 - 0.532i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.25 - 2.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.36 - 4.10i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.498324348493766693944800067397, −8.780583998599399036832625950506, −7.83314042806446249487722887815, −7.15507519547389030007469921308, −5.85512476186877941980405702059, −5.41004257628289697966989246227, −4.83827472323873710084050963412, −3.11414491860508046060402833313, −2.28646225379674858716260343203, −1.24620289268411686628536977595, 1.32168268110450585321917389891, 2.18970749868790524925991550599, 3.23362425270831839185756309645, 4.88573474541397727372743282744, 5.20562105936957420152425932351, 6.11934568746000582772203078471, 6.99110598924581773148524182335, 8.018662845897584938741586200958, 8.633348781933012381369622855450, 9.670555685812472110291765539686

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