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Properties

Label 2-1512-24.11-c1-0-22
Degree $2$
Conductor $1512$
Sign $0.995 + 0.0904i$
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  + (0.743 − 1.20i)2-s + (−0.893 − 1.78i)4-s − 3.69·5-s + i·7-s + (−2.81 − 0.255i)8-s + (−2.74 + 4.44i)10-s + 2.05i·11-s − 4.65i·13-s + (1.20 + 0.743i)14-s + (−2.40 + 3.19i)16-s + 5.73i·17-s + 3.27·19-s + (3.30 + 6.61i)20-s + (2.47 + 1.53i)22-s − 4.45·23-s + ⋯
L(s)  = 1  + (0.525 − 0.850i)2-s + (−0.446 − 0.894i)4-s − 1.65·5-s + 0.377i·7-s + (−0.995 − 0.0904i)8-s + (−0.869 + 1.40i)10-s + 0.620i·11-s − 1.28i·13-s + (0.321 + 0.198i)14-s + (−0.600 + 0.799i)16-s + 1.39i·17-s + 0.750·19-s + (0.738 + 1.47i)20-s + (0.527 + 0.326i)22-s − 0.928·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.995 + 0.0904i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.995 + 0.0904i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.126903228\)
\(L(\frac12)\) \(\approx\) \(1.126903228\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.743 + 1.20i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 + 3.69T + 5T^{2} \)
11 \( 1 - 2.05iT - 11T^{2} \)
13 \( 1 + 4.65iT - 13T^{2} \)
17 \( 1 - 5.73iT - 17T^{2} \)
19 \( 1 - 3.27T + 19T^{2} \)
23 \( 1 + 4.45T + 23T^{2} \)
29 \( 1 - 10.2T + 29T^{2} \)
31 \( 1 - 1.26iT - 31T^{2} \)
37 \( 1 + 3.25iT - 37T^{2} \)
41 \( 1 - 8.72iT - 41T^{2} \)
43 \( 1 + 2.56T + 43T^{2} \)
47 \( 1 - 2.03T + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 + 0.851iT - 59T^{2} \)
61 \( 1 - 2.31iT - 61T^{2} \)
67 \( 1 - 15.4T + 67T^{2} \)
71 \( 1 + 4.44T + 71T^{2} \)
73 \( 1 + 6.97T + 73T^{2} \)
79 \( 1 - 11.6iT - 79T^{2} \)
83 \( 1 - 12.8iT - 83T^{2} \)
89 \( 1 - 2.35iT - 89T^{2} \)
97 \( 1 - 4.02T + 97T^{2} \)
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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

−9.764266578887693701717253648992, −8.353232655581069084936277821788, −8.240608769220977803828245734775, −7.05940424030771231576622013019, −5.99611364696938375809055277879, −5.05492602916620276425644905187, −4.19260284640910916204689172307, −3.50474606200107713051208365453, −2.58626982785891954164537815727, −1.00937721032529094866474497919, 0.48265589304081225113904049534, 2.85422461249113994941876714892, 3.77818163520559026535057137741, 4.39786172650460329457352362756, 5.18988456388382659051522410291, 6.45729521783817052390967461772, 7.12112611882302729084960287890, 7.69633612387027955013189932625, 8.483328774668679820696818273765, 9.108125464379420506345821202409

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