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Properties

Label 2-252-7.5-c8-0-2
Degree $2$
Conductor $252$
Sign $-0.996 + 0.0787i$
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  + (534. + 308. i)5-s + (−750. + 2.28e3i)7-s + (−1.00e4 − 1.73e4i)11-s + 5.49e4i·13-s + (8.22e4 − 4.75e4i)17-s + (5.00e4 + 2.88e4i)19-s + (−2.10e5 + 3.65e5i)23-s + (−4.67e3 − 8.08e3i)25-s − 1.09e6·29-s + (1.89e5 − 1.09e5i)31-s + (−1.10e6 + 9.87e5i)35-s + (−7.70e5 + 1.33e6i)37-s − 3.76e5i·41-s + 1.37e6·43-s + (5.15e6 + 2.97e6i)47-s + ⋯
L(s)  = 1  + (0.855 + 0.493i)5-s + (−0.312 + 0.949i)7-s + (−0.683 − 1.18i)11-s + 1.92i·13-s + (0.985 − 0.568i)17-s + (0.383 + 0.221i)19-s + (−0.753 + 1.30i)23-s + (−0.0119 − 0.0207i)25-s − 1.55·29-s + (0.204 − 0.118i)31-s + (−0.736 + 0.658i)35-s + (−0.411 + 0.712i)37-s − 0.133i·41-s + 0.401·43-s + (1.05 + 0.610i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0787i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.996 + 0.0787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.996 + 0.0787i$
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: \((2,\ 252,\ (\ :4),\ -0.996 + 0.0787i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.8776567783\)
\(L(\frac12)\) \(\approx\) \(0.8776567783\)
\(L(5)\) 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 \( 1 + (750. - 2.28e3i)T \)
good5 \( 1 + (-534. - 308. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (1.00e4 + 1.73e4i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 - 5.49e4iT - 8.15e8T^{2} \)
17 \( 1 + (-8.22e4 + 4.75e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (-5.00e4 - 2.88e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (2.10e5 - 3.65e5i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + 1.09e6T + 5.00e11T^{2} \)
31 \( 1 + (-1.89e5 + 1.09e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (7.70e5 - 1.33e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 + 3.76e5iT - 7.98e12T^{2} \)
43 \( 1 - 1.37e6T + 1.16e13T^{2} \)
47 \( 1 + (-5.15e6 - 2.97e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (6.64e6 + 1.15e7i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (-1.54e7 + 8.89e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (2.14e6 + 1.23e6i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-1.01e7 - 1.75e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 - 1.39e7T + 6.45e14T^{2} \)
73 \( 1 + (1.95e7 - 1.12e7i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (3.41e7 - 5.91e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + 2.02e7iT - 2.25e15T^{2} \)
89 \( 1 + (8.57e7 + 4.95e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 + 1.53e8iT - 7.83e15T^{2} \)
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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

−11.29054414778005000148587228741, −9.864236051993748564539674231346, −9.418401129715082736532527347278, −8.298445781582045201530286904828, −7.05057511261197724989069709188, −5.94010956610196452905589638261, −5.42337666815832141414994001097, −3.67530296637894544560129301629, −2.57720988686710862649780957998, −1.60528455661905149762455479193, 0.17855162442598271233925402780, 1.27544265101498562708443493332, 2.55611047561246053401352289075, 3.86821199095483950673359352088, 5.16754266350831907092941308841, 5.89199615504307771131820005703, 7.34057475077260880497690087494, 7.984769923349842583067887636574, 9.391011708931349986249494107260, 10.25878241484825451019731910387

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