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Properties

Label 2-1512-8.5-c1-0-57
Degree $2$
Conductor $1512$
Sign $0.222 + 0.974i$
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  + (−0.796 − 1.16i)2-s + (−0.730 + 1.86i)4-s + 2.52i·5-s + 7-s + (2.75 − 0.629i)8-s + (2.95 − 2.01i)10-s − 5.70i·11-s + 3.06i·13-s + (−0.796 − 1.16i)14-s + (−2.93 − 2.72i)16-s − 5.49·17-s − 7.28i·19-s + (−4.70 − 1.84i)20-s + (−6.66 + 4.54i)22-s + 0.539·23-s + ⋯
L(s)  = 1  + (−0.563 − 0.826i)2-s + (−0.365 + 0.930i)4-s + 1.12i·5-s + 0.377·7-s + (0.974 − 0.222i)8-s + (0.933 − 0.636i)10-s − 1.71i·11-s + 0.848i·13-s + (−0.212 − 0.312i)14-s + (−0.733 − 0.680i)16-s − 1.33·17-s − 1.67i·19-s + (−1.05 − 0.412i)20-s + (−1.42 + 0.968i)22-s + 0.112·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.222 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.222 + 0.974i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.222 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.094194077\)
\(L(\frac12)\) \(\approx\) \(1.094194077\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.796 + 1.16i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 - 2.52iT - 5T^{2} \)
11 \( 1 + 5.70iT - 11T^{2} \)
13 \( 1 - 3.06iT - 13T^{2} \)
17 \( 1 + 5.49T + 17T^{2} \)
19 \( 1 + 7.28iT - 19T^{2} \)
23 \( 1 - 0.539T + 23T^{2} \)
29 \( 1 + 8.35iT - 29T^{2} \)
31 \( 1 - 6.74T + 31T^{2} \)
37 \( 1 - 10.2iT - 37T^{2} \)
41 \( 1 - 5.58T + 41T^{2} \)
43 \( 1 + 3.98iT - 43T^{2} \)
47 \( 1 - 4.83T + 47T^{2} \)
53 \( 1 + 11.1iT - 53T^{2} \)
59 \( 1 + 10.1iT - 59T^{2} \)
61 \( 1 - 4.14iT - 61T^{2} \)
67 \( 1 - 5.96iT - 67T^{2} \)
71 \( 1 + 4.93T + 71T^{2} \)
73 \( 1 - 8.66T + 73T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 - 5.83iT - 83T^{2} \)
89 \( 1 - 9.28T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−9.278798131800601856176415068772, −8.638656856164769942714555968799, −7.951272789014628332053451509549, −6.80596475699658536173892400000, −6.41330283845947003488497984562, −4.87407129795677358122295914396, −3.96506376482347883460655725602, −2.90221099260962927093151203914, −2.30574816981182049307016916471, −0.62021809850431460543000780227, 1.11385373546413163915113147736, 2.13230383641982383364401506442, 4.17073784680979047853548606801, 4.77719853673917895538691694981, 5.50923006332694506991488708890, 6.46506022288447388623480872241, 7.49999888933208615304171913009, 7.930163011565131154325937295985, 8.935403925751339895068090775107, 9.281505592702573218823038557242

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