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Properties

Label 2-1512-24.11-c1-0-76
Degree $2$
Conductor $1512$
Sign $0.848 - 0.529i$
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  + (1.33 + 0.468i)2-s + (1.56 + 1.25i)4-s + 3.47·5-s + i·7-s + (1.49 + 2.39i)8-s + (4.63 + 1.62i)10-s − 5.11i·11-s − 0.268i·13-s + (−0.468 + 1.33i)14-s + (0.873 + 3.90i)16-s − 1.96i·17-s + 0.909·19-s + (5.41 + 4.33i)20-s + (2.39 − 6.83i)22-s + 5.86·23-s + ⋯
L(s)  = 1  + (0.943 + 0.331i)2-s + (0.780 + 0.625i)4-s + 1.55·5-s + 0.377i·7-s + (0.529 + 0.848i)8-s + (1.46 + 0.514i)10-s − 1.54i·11-s − 0.0744i·13-s + (−0.125 + 0.356i)14-s + (0.218 + 0.975i)16-s − 0.476i·17-s + 0.208·19-s + (1.21 + 0.970i)20-s + (0.511 − 1.45i)22-s + 1.22·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.848 - 0.529i$
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: \((2,\ 1512,\ (\ :1/2),\ 0.848 - 0.529i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.116305898\)
\(L(\frac12)\) \(\approx\) \(4.116305898\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 - 0.468i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 - 3.47T + 5T^{2} \)
11 \( 1 + 5.11iT - 11T^{2} \)
13 \( 1 + 0.268iT - 13T^{2} \)
17 \( 1 + 1.96iT - 17T^{2} \)
19 \( 1 - 0.909T + 19T^{2} \)
23 \( 1 - 5.86T + 23T^{2} \)
29 \( 1 + 7.19T + 29T^{2} \)
31 \( 1 + 4.57iT - 31T^{2} \)
37 \( 1 - 6.98iT - 37T^{2} \)
41 \( 1 - 8.34iT - 41T^{2} \)
43 \( 1 + 9.25T + 43T^{2} \)
47 \( 1 + 6.49T + 47T^{2} \)
53 \( 1 - 4.54T + 53T^{2} \)
59 \( 1 + 3.50iT - 59T^{2} \)
61 \( 1 - 1.96iT - 61T^{2} \)
67 \( 1 + 11.0T + 67T^{2} \)
71 \( 1 - 1.20T + 71T^{2} \)
73 \( 1 + 9.01T + 73T^{2} \)
79 \( 1 + 13.5iT - 79T^{2} \)
83 \( 1 - 4.96iT - 83T^{2} \)
89 \( 1 - 18.2iT - 89T^{2} \)
97 \( 1 - 3.39T + 97T^{2} \)
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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

−9.469458343081970577183492561776, −8.751182430142701884045571644772, −7.87477585500860788956901220668, −6.73593004141362817999735544670, −6.11161010836798464711824775604, −5.49597691782605884287552967330, −4.85153040266836815506887227794, −3.35623083510203171125294479924, −2.71608918823099161911474672788, −1.52058545064240451341137248110, 1.55096056806496058511198963762, 2.13455397141243822557561504487, 3.33099540028257723048932945621, 4.47978653391204552811018480719, 5.23664400951592059701417502878, 5.91192465640685867624292419191, 6.93177401370698144517112617662, 7.29814411489700991968000268775, 8.917597475706600313910005918873, 9.662518434716008743545467252747

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