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Properties

Label 2-1512-24.11-c1-0-30
Degree $2$
Conductor $1512$
Sign $0.998 + 0.0596i$
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  + (−0.682 − 1.23i)2-s + (−1.06 + 1.69i)4-s + 0.381·5-s + i·7-s + (2.82 + 0.168i)8-s + (−0.260 − 0.472i)10-s + 2.13i·11-s − 6.49i·13-s + (1.23 − 0.682i)14-s + (−1.71 − 3.61i)16-s + 6.97i·17-s + 6.19·19-s + (−0.407 + 0.644i)20-s + (2.64 − 1.45i)22-s − 1.82·23-s + ⋯
L(s)  = 1  + (−0.482 − 0.875i)2-s + (−0.534 + 0.845i)4-s + 0.170·5-s + 0.377i·7-s + (0.998 + 0.0596i)8-s + (−0.0823 − 0.149i)10-s + 0.643i·11-s − 1.80i·13-s + (0.331 − 0.182i)14-s + (−0.429 − 0.903i)16-s + 1.69i·17-s + 1.42·19-s + (−0.0910 + 0.144i)20-s + (0.563 − 0.310i)22-s − 0.380·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.998 + 0.0596i$
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: \((2,\ 1512,\ (\ :1/2),\ 0.998 + 0.0596i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.208847560\)
\(L(\frac12)\) \(\approx\) \(1.208847560\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.682 + 1.23i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 - 0.381T + 5T^{2} \)
11 \( 1 - 2.13iT - 11T^{2} \)
13 \( 1 + 6.49iT - 13T^{2} \)
17 \( 1 - 6.97iT - 17T^{2} \)
19 \( 1 - 6.19T + 19T^{2} \)
23 \( 1 + 1.82T + 23T^{2} \)
29 \( 1 - 0.694T + 29T^{2} \)
31 \( 1 + 1.67iT - 31T^{2} \)
37 \( 1 - 8.06iT - 37T^{2} \)
41 \( 1 + 1.31iT - 41T^{2} \)
43 \( 1 - 7.67T + 43T^{2} \)
47 \( 1 - 6.82T + 47T^{2} \)
53 \( 1 + 0.954T + 53T^{2} \)
59 \( 1 - 12.8iT - 59T^{2} \)
61 \( 1 - 12.4iT - 61T^{2} \)
67 \( 1 - 0.634T + 67T^{2} \)
71 \( 1 - 5.79T + 71T^{2} \)
73 \( 1 - 8.13T + 73T^{2} \)
79 \( 1 + 14.1iT - 79T^{2} \)
83 \( 1 + 3.36iT - 83T^{2} \)
89 \( 1 - 15.9iT - 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−9.653190966158130394314896541681, −8.759546244576394173032511123277, −7.927169699715531553052284638822, −7.50144106664256015941977371231, −6.03911710568842206840087349262, −5.31644865066860508616417576754, −4.14567319702089625015300041211, −3.24446863870722181099551011432, −2.28436842325180900425185238391, −1.08960475883513476002637838308, 0.69362604834994675155250236270, 2.08127053360124125321994451896, 3.66104238052359109625584030654, 4.66148129002773469846905249662, 5.48169172818399420057467102988, 6.37195947631199364221612371717, 7.17798922072015444162261142254, 7.67418615239228562406130020144, 8.763002799597283059821659584582, 9.516798409032976169474542295590

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