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Properties

Label 2-252-7.5-c8-0-0
Degree $2$
Conductor $252$
Sign $-0.995 - 0.0931i$
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  + (203. + 117. i)5-s + (1.13e3 − 2.11e3i)7-s + (7.47e3 + 1.29e4i)11-s + 3.96e4i·13-s + (−8.46e4 + 4.88e4i)17-s + (−1.77e5 − 1.02e5i)19-s + (1.85e5 − 3.20e5i)23-s + (−1.67e5 − 2.90e5i)25-s − 1.17e5·29-s + (4.37e5 − 2.52e5i)31-s + (4.78e5 − 2.98e5i)35-s + (−9.86e5 + 1.70e6i)37-s + 3.25e6i·41-s − 4.13e6·43-s + (4.79e6 + 2.76e6i)47-s + ⋯
L(s)  = 1  + (0.325 + 0.187i)5-s + (0.470 − 0.882i)7-s + (0.510 + 0.884i)11-s + 1.38i·13-s + (−1.01 + 0.585i)17-s + (−1.36 − 0.787i)19-s + (0.662 − 1.14i)23-s + (−0.429 − 0.743i)25-s − 0.165·29-s + (0.474 − 0.273i)31-s + (0.318 − 0.198i)35-s + (−0.526 + 0.911i)37-s + 1.15i·41-s − 1.20·43-s + (0.982 + 0.567i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0931i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.995 - 0.0931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.995 - 0.0931i$
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: \((2,\ 252,\ (\ :4),\ -0.995 - 0.0931i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.2364175435\)
\(L(\frac12)\) \(\approx\) \(0.2364175435\)
\(L(5)\) 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 \( 1 + (-1.13e3 + 2.11e3i)T \)
good5 \( 1 + (-203. - 117. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (-7.47e3 - 1.29e4i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 - 3.96e4iT - 8.15e8T^{2} \)
17 \( 1 + (8.46e4 - 4.88e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (1.77e5 + 1.02e5i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (-1.85e5 + 3.20e5i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + 1.17e5T + 5.00e11T^{2} \)
31 \( 1 + (-4.37e5 + 2.52e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (9.86e5 - 1.70e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 - 3.25e6iT - 7.98e12T^{2} \)
43 \( 1 + 4.13e6T + 1.16e13T^{2} \)
47 \( 1 + (-4.79e6 - 2.76e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (-3.39e6 - 5.87e6i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (-1.91e7 + 1.10e7i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (3.97e6 + 2.29e6i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (1.86e7 + 3.22e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 + 4.19e7T + 6.45e14T^{2} \)
73 \( 1 + (-1.08e7 + 6.27e6i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (2.01e7 - 3.48e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 - 5.31e6iT - 2.25e15T^{2} \)
89 \( 1 + (6.45e7 + 3.72e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 - 3.76e7iT - 7.83e15T^{2} \)
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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

−10.98893493711067351344512369005, −10.22359456078782251580788379448, −9.150990799271794868321314683742, −8.255157163326787737841394297367, −6.75457130848951671569241922598, −6.59291412315165731128247395409, −4.54539603837038583108683020032, −4.27272886466162636917379273712, −2.38091674890544825921570039176, −1.49113976833009532383962044013, 0.04705126685352096521971605245, 1.43522021556499164068909441824, 2.58694429590534010097574615249, 3.82638041888940490941711263019, 5.32698091328965356546085306405, 5.84765169676099436409846957903, 7.19798335999401786332734507183, 8.504873696323326119212269206425, 8.912439709330837054863526285936, 10.20129037805486965488850869681

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