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Properties

Label 2-1512-21.17-c1-0-18
Degree $2$
Conductor $1512$
Sign $0.562 + 0.826i$
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  + (−0.310 − 0.537i)5-s + (−1.44 + 2.21i)7-s + (−1.91 − 1.10i)11-s − 0.963i·13-s + (0.653 − 1.13i)17-s + (−1.57 + 0.911i)19-s + (5.27 − 3.04i)23-s + (2.30 − 3.99i)25-s − 6.68i·29-s + (3.48 + 2.01i)31-s + (1.63 + 0.0859i)35-s + (0.165 + 0.286i)37-s + 11.7·41-s − 2.18·43-s + (−1.27 − 2.20i)47-s + ⋯
L(s)  = 1  + (−0.138 − 0.240i)5-s + (−0.544 + 0.838i)7-s + (−0.577 − 0.333i)11-s − 0.267i·13-s + (0.158 − 0.274i)17-s + (−0.362 + 0.209i)19-s + (1.10 − 0.635i)23-s + (0.461 − 0.799i)25-s − 1.24i·29-s + (0.626 + 0.361i)31-s + (0.277 + 0.0145i)35-s + (0.0271 + 0.0470i)37-s + 1.83·41-s − 0.333·43-s + (−0.185 − 0.321i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.562 + 0.826i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.562 + 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.280965694\)
\(L(\frac12)\) \(\approx\) \(1.280965694\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.44 - 2.21i)T \)
good5 \( 1 + (0.310 + 0.537i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.91 + 1.10i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.963iT - 13T^{2} \)
17 \( 1 + (-0.653 + 1.13i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.57 - 0.911i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.27 + 3.04i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.68iT - 29T^{2} \)
31 \( 1 + (-3.48 - 2.01i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.165 - 0.286i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.7T + 41T^{2} \)
43 \( 1 + 2.18T + 43T^{2} \)
47 \( 1 + (1.27 + 2.20i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.985 - 0.569i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.20 + 10.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.603 + 0.348i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.61 - 6.25i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.40iT - 71T^{2} \)
73 \( 1 + (2.17 + 1.25i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.52 - 7.83i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.85T + 83T^{2} \)
89 \( 1 + (4.20 + 7.29i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.4iT - 97T^{2} \)
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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

−9.308191977157950065148828971117, −8.512390346242674387545149647083, −7.956868164028889887016063385595, −6.81721673417968216360257163884, −6.06505872703600359341276182556, −5.25125429954569123120304246373, −4.35360590919092273018400808219, −3.09150311924977690435047122635, −2.37546869996277260073052503565, −0.58512265967090040623169241082, 1.13568785292751868095358800419, 2.66823649793337296276859810215, 3.56539094117109623514778721157, 4.51653513764435873193846852627, 5.44286516128832762601182208795, 6.53065945742351604049675765866, 7.19914196339591648034661148173, 7.80077840701541208751959642316, 8.939731305351096913318369641953, 9.560050896761407919446574184718

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