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Properties

Label 2-252-21.17-c7-0-3
Degree $2$
Conductor $252$
Sign $0.888 - 0.458i$
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  + (−275. − 477. i)5-s + (877. − 230. i)7-s + (1.25e3 + 722. i)11-s + 1.24e4i·13-s + (8.65e3 − 1.49e4i)17-s + (−2.25e4 + 1.30e4i)19-s + (−6.79e4 + 3.92e4i)23-s + (−1.12e5 + 1.95e5i)25-s − 3.35e4i·29-s + (1.96e5 + 1.13e5i)31-s + (−3.52e5 − 3.55e5i)35-s + (−1.75e5 − 3.03e5i)37-s + 1.09e5·41-s − 1.10e5·43-s + (5.13e5 + 8.90e5i)47-s + ⋯
L(s)  = 1  + (−0.985 − 1.70i)5-s + (0.967 − 0.254i)7-s + (0.283 + 0.163i)11-s + 1.57i·13-s + (0.427 − 0.740i)17-s + (−0.754 + 0.435i)19-s + (−1.16 + 0.672i)23-s + (−1.44 + 2.50i)25-s − 0.255i·29-s + (1.18 + 0.683i)31-s + (−1.38 − 1.40i)35-s + (−0.568 − 0.984i)37-s + 0.248·41-s − 0.211·43-s + (0.722 + 1.25i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.888 - 0.458i$
Analytic conductor: \(78.7210\)
Root analytic conductor: \(8.87248\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :7/2),\ 0.888 - 0.458i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.462484399\)
\(L(\frac12)\) \(\approx\) \(1.462484399\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-877. + 230. i)T \)
good5 \( 1 + (275. + 477. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
11 \( 1 + (-1.25e3 - 722. i)T + (9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 - 1.24e4iT - 6.27e7T^{2} \)
17 \( 1 + (-8.65e3 + 1.49e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (2.25e4 - 1.30e4i)T + (4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 + (6.79e4 - 3.92e4i)T + (1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 + 3.35e4iT - 1.72e10T^{2} \)
31 \( 1 + (-1.96e5 - 1.13e5i)T + (1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 + (1.75e5 + 3.03e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 - 1.09e5T + 1.94e11T^{2} \)
43 \( 1 + 1.10e5T + 2.71e11T^{2} \)
47 \( 1 + (-5.13e5 - 8.90e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (-8.68e5 - 5.01e5i)T + (5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (1.13e5 - 1.95e5i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-3.40e5 + 1.96e5i)T + (1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (3.93e5 - 6.81e5i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 - 2.32e6iT - 9.09e12T^{2} \)
73 \( 1 + (-1.33e6 - 7.71e5i)T + (5.52e12 + 9.56e12i)T^{2} \)
79 \( 1 + (-3.43e6 - 5.95e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + 6.19e6T + 2.71e13T^{2} \)
89 \( 1 + (4.21e6 + 7.29e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 - 7.20e6iT - 8.07e13T^{2} \)
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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

−11.19520849858502517032775046567, −9.676752529181927530883175933515, −8.785454839525135595945938433133, −8.090799632316520467142286945360, −7.17289380611051265225240266463, −5.54883142114836322484415375681, −4.43117870047202521226013095372, −4.07220059943959991860344210315, −1.82327209820503143712834361654, −0.934087369537489235001450806867, 0.41950839292759119189691664231, 2.25189105018520391432076749621, 3.27658451381418353450624255398, 4.30529844128973499830518323978, 5.82761225807155728546668295170, 6.80352795388277502565551097492, 7.973706296411023390611856818983, 8.288845843152609582668395510009, 10.29162214787882992001608773720, 10.60363475328915229077830011532

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