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Properties

Label 2-1512-24.11-c1-0-68
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

\(L(1)\) \(\approx\) \(1.622423637\)
\(L(\frac12)\) \(\approx\) \(1.622423637\)
\(L(\frac{3}{2})\) 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

−9.369197580231255809955161607155, −8.406534845339396718377553960235, −7.75486590496993009876191807517, −7.17023078821037700487242416635, −5.99288102888742631991410728475, −5.43426584443291279766164186697, −4.72707263099232482182258245485, −3.30744465515711850370361601374, −2.90610320660252458216815849101, −0.58398285230422940033533606900, 1.37452360577414228852618300648, 2.24350433331226480516481727935, 3.71037666247311254537484759695, 4.12478286458739187013957628803, 5.12718581517040159257974936567, 6.10429639263476033836573382525, 6.95892319649665456696302216196, 7.900368365429936429967753927344, 9.060295086963845377513523449426, 9.517011553202555155034398269230

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