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Properties

Label 2-252-7.3-c8-0-8
Degree $2$
Conductor $252$
Sign $-0.302 - 0.953i$
Analytic cond. $102.659$
Root an. cond. $10.1320$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.01e3 + 584. i)5-s + (−1.82e3 + 1.55e3i)7-s + (1.30e4 − 2.26e4i)11-s + 2.85e4i·13-s + (1.80e4 + 1.04e4i)17-s + (4.29e4 − 2.48e4i)19-s + (−1.70e5 − 2.96e5i)23-s + (4.87e5 − 8.43e5i)25-s + 1.12e6·29-s + (1.47e6 + 8.51e5i)31-s + (9.43e5 − 2.64e6i)35-s + (3.20e5 + 5.54e5i)37-s − 1.34e6i·41-s − 1.80e6·43-s + (−7.88e6 + 4.55e6i)47-s + ⋯
L(s)  = 1  + (−1.61 + 0.934i)5-s + (−0.762 + 0.647i)7-s + (0.893 − 1.54i)11-s + 1.00i·13-s + (0.215 + 0.124i)17-s + (0.329 − 0.190i)19-s + (−0.610 − 1.05i)23-s + (1.24 − 2.16i)25-s + 1.59·29-s + (1.59 + 0.922i)31-s + (0.628 − 1.76i)35-s + (0.170 + 0.295i)37-s − 0.476i·41-s − 0.526·43-s + (−1.61 + 0.933i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.302 - 0.953i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.302 - 0.953i$
Analytic conductor: \(102.659\)
Root analytic conductor: \(10.1320\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :4),\ -0.302 - 0.953i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.000679539\)
\(L(\frac12)\) \(\approx\) \(1.000679539\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (1.82e3 - 1.55e3i)T \)
good5 \( 1 + (1.01e3 - 584. i)T + (1.95e5 - 3.38e5i)T^{2} \)
11 \( 1 + (-1.30e4 + 2.26e4i)T + (-1.07e8 - 1.85e8i)T^{2} \)
13 \( 1 - 2.85e4iT - 8.15e8T^{2} \)
17 \( 1 + (-1.80e4 - 1.04e4i)T + (3.48e9 + 6.04e9i)T^{2} \)
19 \( 1 + (-4.29e4 + 2.48e4i)T + (8.49e9 - 1.47e10i)T^{2} \)
23 \( 1 + (1.70e5 + 2.96e5i)T + (-3.91e10 + 6.78e10i)T^{2} \)
29 \( 1 - 1.12e6T + 5.00e11T^{2} \)
31 \( 1 + (-1.47e6 - 8.51e5i)T + (4.26e11 + 7.38e11i)T^{2} \)
37 \( 1 + (-3.20e5 - 5.54e5i)T + (-1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 + 1.34e6iT - 7.98e12T^{2} \)
43 \( 1 + 1.80e6T + 1.16e13T^{2} \)
47 \( 1 + (7.88e6 - 4.55e6i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (-1.82e6 + 3.16e6i)T + (-3.11e13 - 5.39e13i)T^{2} \)
59 \( 1 + (9.97e6 + 5.76e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (8.77e6 - 5.06e6i)T + (9.58e13 - 1.66e14i)T^{2} \)
67 \( 1 + (8.86e6 - 1.53e7i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 - 3.79e7T + 6.45e14T^{2} \)
73 \( 1 + (-2.94e7 - 1.70e7i)T + (4.03e14 + 6.98e14i)T^{2} \)
79 \( 1 + (9.66e6 + 1.67e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 + 9.60e5iT - 2.25e15T^{2} \)
89 \( 1 + (5.62e7 - 3.24e7i)T + (1.96e15 - 3.40e15i)T^{2} \)
97 \( 1 + 2.66e7iT - 7.83e15T^{2} \)
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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

−11.12816905985728361247732732920, −10.04505792770876363422263346560, −8.699364189756359127243634298631, −8.168084231064319875320353816098, −6.62628527477083953926836825417, −6.42294381135335505509019756476, −4.51219973816452021828709287801, −3.46111980974983355889623071287, −2.81886658554225651002083547352, −0.793496739176099781290516034171, 0.33790150604070740452926394619, 1.25623677442934416420383848760, 3.23135019354499175989483460087, 4.13518107534772555964142103888, 4.88669186666951556424458993064, 6.53589467748263018690651314216, 7.57481025680738756419644960328, 8.137913739252835819595591650467, 9.464760539892855419589341430662, 10.15601242243652746321730662323

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