Properties

Label 2-252-12.11-c7-0-28
Degree $2$
Conductor $252$
Sign $-0.125 - 0.992i$
Analytic cond. $78.7210$
Root an. cond. $8.87248$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.56 + 10.3i)2-s + (−86.3 − 94.4i)4-s + 0.594i·5-s + 343i·7-s + (1.37e3 − 463. i)8-s + (−6.15 − 2.71i)10-s + 5.43e3·11-s − 1.25e4·13-s + (−3.55e3 − 1.56e3i)14-s + (−1.45e3 + 1.63e4i)16-s + 2.10e4i·17-s − 3.52e4i·19-s + (56.1 − 51.3i)20-s + (−2.48e4 + 5.62e4i)22-s − 3.27e4·23-s + ⋯
L(s)  = 1  + (−0.403 + 0.915i)2-s + (−0.674 − 0.737i)4-s + 0.00212i·5-s + 0.377i·7-s + (0.947 − 0.320i)8-s + (−0.00194 − 0.000857i)10-s + 1.23·11-s − 1.58·13-s + (−0.345 − 0.152i)14-s + (−0.0888 + 0.996i)16-s + 1.04i·17-s − 1.18i·19-s + (0.00156 − 0.00143i)20-s + (−0.496 + 1.12i)22-s − 0.561·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.125 - 0.992i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.125 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.125 - 0.992i$
Analytic conductor: \(78.7210\)
Root analytic conductor: \(8.87248\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :7/2),\ -0.125 - 0.992i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.363455057\)
\(L(\frac12)\) \(\approx\) \(1.363455057\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.56 - 10.3i)T \)
3 \( 1 \)
7 \( 1 - 343iT \)
good5 \( 1 - 0.594iT - 7.81e4T^{2} \)
11 \( 1 - 5.43e3T + 1.94e7T^{2} \)
13 \( 1 + 1.25e4T + 6.27e7T^{2} \)
17 \( 1 - 2.10e4iT - 4.10e8T^{2} \)
19 \( 1 + 3.52e4iT - 8.93e8T^{2} \)
23 \( 1 + 3.27e4T + 3.40e9T^{2} \)
29 \( 1 + 9.04e4iT - 1.72e10T^{2} \)
31 \( 1 + 4.95e4iT - 2.75e10T^{2} \)
37 \( 1 - 1.52e5T + 9.49e10T^{2} \)
41 \( 1 + 7.66e5iT - 1.94e11T^{2} \)
43 \( 1 - 8.60e5iT - 2.71e11T^{2} \)
47 \( 1 - 1.57e5T + 5.06e11T^{2} \)
53 \( 1 - 3.58e5iT - 1.17e12T^{2} \)
59 \( 1 - 3.53e5T + 2.48e12T^{2} \)
61 \( 1 - 2.00e6T + 3.14e12T^{2} \)
67 \( 1 + 2.30e6iT - 6.06e12T^{2} \)
71 \( 1 + 2.65e6T + 9.09e12T^{2} \)
73 \( 1 - 4.65e6T + 1.10e13T^{2} \)
79 \( 1 - 8.27e6iT - 1.92e13T^{2} \)
83 \( 1 - 1.63e6T + 2.71e13T^{2} \)
89 \( 1 - 9.65e6iT - 4.42e13T^{2} \)
97 \( 1 - 1.50e7T + 8.07e13T^{2} \)
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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

−10.87987981248700216806014991598, −9.730505628710589210590868402884, −9.125809866451728762972985457350, −8.105251224286212155096260757962, −7.04789927995517664290979754790, −6.26052308096240774481100872670, −5.08602453638537683610571312869, −4.08816224628170491684240328099, −2.25484772325735711864485512480, −0.795897240281778181450685995977, 0.51974670825044873636529831521, 1.68277976824103378641224870557, 2.94018564542864169517600921897, 4.08893010219582809512354655702, 5.09256136209883452801401386712, 6.82130314226408963903649369882, 7.69794207759782975207300888686, 8.880361812512398338617698057521, 9.722831820281632531409769360123, 10.39707923157892630572007047706

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