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Properties

Label 2-1512-8.5-c1-0-23
Degree $2$
Conductor $1512$
Sign $-0.951 - 0.309i$
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  + (1.14 + 0.831i)2-s + (0.618 + 1.90i)4-s + 1.60i·5-s − 7-s + (−0.874 + 2.68i)8-s + (−1.33 + 1.83i)10-s − 0.0549i·11-s + 3.75i·13-s + (−1.14 − 0.831i)14-s + (−3.23 + 2.35i)16-s + 3.16·17-s + 0.726i·19-s + (−3.05 + 0.993i)20-s + (0.0456 − 0.0628i)22-s − 7.77·23-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + 0.718i·5-s − 0.377·7-s + (−0.309 + 0.951i)8-s + (−0.422 + 0.581i)10-s − 0.0165i·11-s + 1.04i·13-s + (−0.305 − 0.222i)14-s + (−0.809 + 0.587i)16-s + 0.766·17-s + 0.166i·19-s + (−0.683 + 0.222i)20-s + (0.00973 − 0.0133i)22-s − 1.62·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 - 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.951 - 0.309i$
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: \((2,\ 1512,\ (\ :1/2),\ -0.951 - 0.309i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.091740669\)
\(L(\frac12)\) \(\approx\) \(2.091740669\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.14 - 0.831i)T \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 1.60iT - 5T^{2} \)
11 \( 1 + 0.0549iT - 11T^{2} \)
13 \( 1 - 3.75iT - 13T^{2} \)
17 \( 1 - 3.16T + 17T^{2} \)
19 \( 1 - 0.726iT - 19T^{2} \)
23 \( 1 + 7.77T + 23T^{2} \)
29 \( 1 + 2.75iT - 29T^{2} \)
31 \( 1 + 2.14T + 31T^{2} \)
37 \( 1 - 0.600iT - 37T^{2} \)
41 \( 1 + 4.53T + 41T^{2} \)
43 \( 1 - 6.85iT - 43T^{2} \)
47 \( 1 + 0.744T + 47T^{2} \)
53 \( 1 - 10.5iT - 53T^{2} \)
59 \( 1 + 2.58iT - 59T^{2} \)
61 \( 1 + 9.80iT - 61T^{2} \)
67 \( 1 - 12.2iT - 67T^{2} \)
71 \( 1 + 9.19T + 71T^{2} \)
73 \( 1 - 3.85T + 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + 13.1iT - 83T^{2} \)
89 \( 1 - 7.90T + 89T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−9.834272177880554168780587927041, −8.920733505899872571580214083776, −7.944331482591118273159409692538, −7.30400023694124053281738829402, −6.41043651854459177056279431520, −5.98665717110743478043649426190, −4.82225146680140789770851385928, −3.91136304227327961985781004019, −3.12313159997206720691045312923, −2.02982826274032741202776854043, 0.60138759301964697382166167842, 1.90194272431869611927979444135, 3.13624895830560757320479260816, 3.88538034390936298404842370197, 4.99121934468143679895901703410, 5.56014614932729485506458855467, 6.40599065542695351056757162350, 7.45151951368447566159816078370, 8.400470078929966726581841554093, 9.285153193746074448766801428592

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