Properties

Label 2-252-7.5-c8-0-20
Degree $2$
Conductor $252$
Sign $0.500 - 0.865i$
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  + (939. + 542. i)5-s + (1.45e3 + 1.91e3i)7-s + (8.43e3 + 1.46e4i)11-s − 9.95e3i·13-s + (9.17e4 − 5.29e4i)17-s + (1.47e5 + 8.52e4i)19-s + (1.05e5 − 1.83e5i)23-s + (3.93e5 + 6.81e5i)25-s + 5.67e5·29-s + (5.49e5 − 3.17e5i)31-s + (3.26e5 + 2.58e6i)35-s + (−1.07e6 + 1.86e6i)37-s − 5.45e6i·41-s + 5.56e6·43-s + (−2.76e6 − 1.59e6i)47-s + ⋯
L(s)  = 1  + (1.50 + 0.867i)5-s + (0.604 + 0.796i)7-s + (0.576 + 0.997i)11-s − 0.348i·13-s + (1.09 − 0.634i)17-s + (1.13 + 0.654i)19-s + (0.377 − 0.654i)23-s + (1.00 + 1.74i)25-s + 0.802·29-s + (0.594 − 0.343i)31-s + (0.217 + 1.72i)35-s + (−0.574 + 0.995i)37-s − 1.93i·41-s + 1.62·43-s + (−0.567 − 0.327i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.500 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.500 - 0.865i$
Analytic conductor: \(102.659\)
Root analytic conductor: \(10.1320\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :4),\ 0.500 - 0.865i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(4.060108407\)
\(L(\frac12)\) \(\approx\) \(4.060108407\)
\(L(5)\) 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 \( 1 + (-1.45e3 - 1.91e3i)T \)
good5 \( 1 + (-939. - 542. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (-8.43e3 - 1.46e4i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 + 9.95e3iT - 8.15e8T^{2} \)
17 \( 1 + (-9.17e4 + 5.29e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (-1.47e5 - 8.52e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (-1.05e5 + 1.83e5i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 - 5.67e5T + 5.00e11T^{2} \)
31 \( 1 + (-5.49e5 + 3.17e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (1.07e6 - 1.86e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 + 5.45e6iT - 7.98e12T^{2} \)
43 \( 1 - 5.56e6T + 1.16e13T^{2} \)
47 \( 1 + (2.76e6 + 1.59e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (4.38e6 + 7.59e6i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (-8.08e6 + 4.66e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (2.13e7 + 1.23e7i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (6.91e6 + 1.19e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 + 1.19e7T + 6.45e14T^{2} \)
73 \( 1 + (1.83e7 - 1.05e7i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (8.61e6 - 1.49e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + 6.82e7iT - 2.25e15T^{2} \)
89 \( 1 + (-7.87e7 - 4.54e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 - 3.95e7iT - 7.83e15T^{2} \)
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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

−10.53231630232054397507288761672, −9.849270495511408415396474014229, −9.116697364451511481346346328997, −7.74844318812599811285808905914, −6.70574397090041554070533305848, −5.73664860366408137207616971485, −4.95278209980114880534473916150, −3.14248589806732043969734613261, −2.20267261059337287145318131528, −1.25377842846062564738723059038, 1.09664468314336194474678021504, 1.30283695604044385994111740694, 2.99431309726404711906293801424, 4.44155769844120428526951100255, 5.44668111435420916320974216379, 6.24044401934340774870992067594, 7.56943383065117196892042335852, 8.703894687871374621719041501446, 9.465366761965029540285608051309, 10.33234822078549938722694323718

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