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Properties

Label 2-1512-63.4-c1-0-17
Degree $2$
Conductor $1512$
Sign $0.993 + 0.116i$
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  + 1.83·5-s + (2.45 + 0.997i)7-s + 3.09·11-s + (2.40 − 4.16i)13-s + (−1.87 + 3.24i)17-s + (−2.71 − 4.70i)19-s + 7.95·23-s − 1.62·25-s + (0.325 + 0.563i)29-s + (−0.518 − 0.897i)31-s + (4.50 + 1.83i)35-s + (0.873 + 1.51i)37-s + (−2.52 + 4.36i)41-s + (−6.09 − 10.5i)43-s + (−2.30 + 3.99i)47-s + ⋯
L(s)  = 1  + 0.821·5-s + (0.926 + 0.376i)7-s + 0.933·11-s + (0.666 − 1.15i)13-s + (−0.453 + 0.786i)17-s + (−0.622 − 1.07i)19-s + 1.65·23-s − 0.325·25-s + (0.0604 + 0.104i)29-s + (−0.0930 − 0.161i)31-s + (0.760 + 0.309i)35-s + (0.143 + 0.248i)37-s + (−0.393 + 0.682i)41-s + (−0.929 − 1.61i)43-s + (−0.336 + 0.582i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.993 + 0.116i$
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: \((2,\ 1512,\ (\ :1/2),\ 0.993 + 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.378191654\)
\(L(\frac12)\) \(\approx\) \(2.378191654\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.45 - 0.997i)T \)
good5 \( 1 - 1.83T + 5T^{2} \)
11 \( 1 - 3.09T + 11T^{2} \)
13 \( 1 + (-2.40 + 4.16i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.87 - 3.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.71 + 4.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.95T + 23T^{2} \)
29 \( 1 + (-0.325 - 0.563i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.518 + 0.897i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.873 - 1.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.52 - 4.36i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.09 + 10.5i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.30 - 3.99i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.55 - 7.88i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.89 + 5.02i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.40 + 4.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.23 - 12.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.00T + 71T^{2} \)
73 \( 1 + (1.81 - 3.14i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.17 + 12.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.83 + 6.63i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.76 - 9.99i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.04 + 1.80i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.273651638298987065519613838365, −8.738028399123264908779571804021, −8.048359241521953594824981648614, −6.89244544109446159758900376739, −6.17518835488188662528342775128, −5.34570926495156232868097372980, −4.55754514260688268350773729321, −3.35434166339659995964640928660, −2.18889301954744984220079693472, −1.17678365211628822069884006628, 1.32334158711364736331110035450, 2.06293422155306714044827582864, 3.58036407060895930676106025383, 4.48408821960058419522709291809, 5.29045588187921729558885384051, 6.41561449654117168843358466188, 6.84446556645185606096067086384, 7.970678175423560288647536941901, 8.831788935804797458979185683910, 9.378823395368841682704904766649

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