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Properties

Label 2-252-7.5-c8-0-18
Degree $2$
Conductor $252$
Sign $0.497 + 0.867i$
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  + (616. + 355. i)5-s + (−2.38e3 + 297. i)7-s + (1.14e3 + 1.97e3i)11-s + 8.31e3i·13-s + (6.32e4 − 3.65e4i)17-s + (−2.18e5 − 1.26e5i)19-s + (−1.42e5 + 2.46e5i)23-s + (5.77e4 + 9.99e4i)25-s + 1.05e5·29-s + (7.43e5 − 4.29e5i)31-s + (−1.57e6 − 6.63e5i)35-s + (1.41e6 − 2.45e6i)37-s − 1.75e6i·41-s − 1.66e6·43-s + (−3.13e6 − 1.81e6i)47-s + ⋯
L(s)  = 1  + (0.985 + 0.569i)5-s + (−0.992 + 0.124i)7-s + (0.0779 + 0.135i)11-s + 0.291i·13-s + (0.757 − 0.437i)17-s + (−1.67 − 0.966i)19-s + (−0.508 + 0.880i)23-s + (0.147 + 0.255i)25-s + 0.148·29-s + (0.805 − 0.464i)31-s + (−1.04 − 0.442i)35-s + (0.757 − 1.31i)37-s − 0.621i·41-s − 0.485·43-s + (−0.642 − 0.371i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.497 + 0.867i$
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: \((2,\ 252,\ (\ :4),\ 0.497 + 0.867i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.632254701\)
\(L(\frac12)\) \(\approx\) \(1.632254701\)
\(L(5)\) 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 \( 1 + (2.38e3 - 297. i)T \)
good5 \( 1 + (-616. - 355. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (-1.14e3 - 1.97e3i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 - 8.31e3iT - 8.15e8T^{2} \)
17 \( 1 + (-6.32e4 + 3.65e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (2.18e5 + 1.26e5i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (1.42e5 - 2.46e5i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 - 1.05e5T + 5.00e11T^{2} \)
31 \( 1 + (-7.43e5 + 4.29e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-1.41e6 + 2.45e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 + 1.75e6iT - 7.98e12T^{2} \)
43 \( 1 + 1.66e6T + 1.16e13T^{2} \)
47 \( 1 + (3.13e6 + 1.81e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (-5.68e6 - 9.84e6i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (-1.15e7 + 6.68e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (1.24e7 + 7.17e6i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-1.55e7 - 2.68e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 + 3.33e7T + 6.45e14T^{2} \)
73 \( 1 + (7.19e6 - 4.15e6i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (-3.40e6 + 5.90e6i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 - 4.05e7iT - 2.25e15T^{2} \)
89 \( 1 + (-3.65e7 - 2.11e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 + 9.29e7iT - 7.83e15T^{2} \)
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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

−10.22318068982680690066930936961, −9.673812164735819421719563076457, −8.728493655437156370697790299731, −7.26813010131811440094709265881, −6.39714893610213576446460127651, −5.67282678913162576539471107415, −4.17701660620439936676435484926, −2.86692380998017215458874429262, −2.01529870631166911087227077164, −0.38381738638174122102184906683, 0.967425745179928370431818980605, 2.14810806581232798016572976291, 3.42688733705332132899333874244, 4.68754903886193247902749922625, 6.01817414218697573256706025639, 6.42121708326480671501457033496, 8.043151354944021710017089773631, 8.875059953352849335935963625281, 10.10410421585734600697880971156, 10.24939101017014427004373180669

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