Properties

Label 2-252-21.11-c8-0-1
Degree $2$
Conductor $252$
Sign $-0.976 - 0.213i$
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  + (−23.4 + 13.5i)5-s + (−2.30e3 + 663. i)7-s + (6.53e3 + 3.77e3i)11-s + 1.07e4·13-s + (−4.87e4 − 2.81e4i)17-s + (5.54e4 + 9.60e4i)19-s + (4.09e5 − 2.36e5i)23-s + (−1.94e5 + 3.37e5i)25-s + 3.86e5i·29-s + (7.68e5 − 1.33e6i)31-s + (4.50e4 − 4.67e4i)35-s + (7.92e5 + 1.37e6i)37-s − 7.94e5i·41-s − 6.19e6·43-s + (−6.61e6 + 3.81e6i)47-s + ⋯
L(s)  = 1  + (−0.0374 + 0.0216i)5-s + (−0.961 + 0.276i)7-s + (0.446 + 0.257i)11-s + 0.375·13-s + (−0.584 − 0.337i)17-s + (0.425 + 0.737i)19-s + (1.46 − 0.845i)23-s + (−0.499 + 0.864i)25-s + 0.545i·29-s + (0.831 − 1.44i)31-s + (0.0300 − 0.0311i)35-s + (0.423 + 0.732i)37-s − 0.281i·41-s − 1.81·43-s + (−1.35 + 0.782i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.213i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.976 - 0.213i$
Analytic conductor: \(102.659\)
Root analytic conductor: \(10.1320\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :4),\ -0.976 - 0.213i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.4243588684\)
\(L(\frac12)\) \(\approx\) \(0.4243588684\)
\(L(5)\) 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 \( 1 + (2.30e3 - 663. i)T \)
good5 \( 1 + (23.4 - 13.5i)T + (1.95e5 - 3.38e5i)T^{2} \)
11 \( 1 + (-6.53e3 - 3.77e3i)T + (1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 - 1.07e4T + 8.15e8T^{2} \)
17 \( 1 + (4.87e4 + 2.81e4i)T + (3.48e9 + 6.04e9i)T^{2} \)
19 \( 1 + (-5.54e4 - 9.60e4i)T + (-8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (-4.09e5 + 2.36e5i)T + (3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 - 3.86e5iT - 5.00e11T^{2} \)
31 \( 1 + (-7.68e5 + 1.33e6i)T + (-4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-7.92e5 - 1.37e6i)T + (-1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 + 7.94e5iT - 7.98e12T^{2} \)
43 \( 1 + 6.19e6T + 1.16e13T^{2} \)
47 \( 1 + (6.61e6 - 3.81e6i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (2.25e6 + 1.30e6i)T + (3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (-1.13e7 - 6.54e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (2.09e6 + 3.62e6i)T + (-9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (3.05e6 - 5.28e6i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 + 2.56e7iT - 6.45e14T^{2} \)
73 \( 1 + (-4.61e5 + 7.99e5i)T + (-4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (9.41e5 + 1.63e6i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 - 8.96e7iT - 2.25e15T^{2} \)
89 \( 1 + (-2.07e7 + 1.19e7i)T + (1.96e15 - 3.40e15i)T^{2} \)
97 \( 1 + 1.05e7T + 7.83e15T^{2} \)
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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

−11.13011597983239896789034829321, −9.926283559661575087279248438359, −9.272833772956084650970512845787, −8.228242141980623258388375296634, −6.94945556266265804040977981721, −6.23444538669880977565792384342, −4.99092700997170275190529686346, −3.71180299944181659181158935247, −2.70964759211176121430534127126, −1.26278074368400601262134243419, 0.098199234747110807870854009100, 1.26977273848639357372761024822, 2.84863846388961636117899395442, 3.78058650728363423063740019012, 5.04006469937611676823302023788, 6.36180165212671472028856991335, 6.97628617454319621634581825995, 8.335822120679550817509211570015, 9.240930865367761585826946597016, 10.10753975045677445185359692889

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