Properties

Label 2-252-3.2-c8-0-15
Degree $2$
Conductor $252$
Sign $-0.577 - 0.816i$
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  − 941. i·5-s + 907.·7-s − 2.49e4i·11-s − 5.39e4·13-s − 1.26e5i·17-s + 2.19e3·19-s + 4.99e5i·23-s − 4.95e5·25-s − 4.71e5i·29-s − 1.20e6·31-s − 8.54e5i·35-s − 9.05e4·37-s − 5.01e6i·41-s + 1.25e6·43-s + 4.26e6i·47-s + ⋯
L(s)  = 1  − 1.50i·5-s + 0.377·7-s − 1.70i·11-s − 1.88·13-s − 1.51i·17-s + 0.0168·19-s + 1.78i·23-s − 1.26·25-s − 0.666i·29-s − 1.30·31-s − 0.569i·35-s − 0.0483·37-s − 1.77i·41-s + 0.367·43-s + 0.873i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(102.659\)
Root analytic conductor: \(10.1320\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :4),\ -0.577 - 0.816i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.7319113571\)
\(L(\frac12)\) \(\approx\) \(0.7319113571\)
\(L(5)\) 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 \( 1 - 907.T \)
good5 \( 1 + 941. iT - 3.90e5T^{2} \)
11 \( 1 + 2.49e4iT - 2.14e8T^{2} \)
13 \( 1 + 5.39e4T + 8.15e8T^{2} \)
17 \( 1 + 1.26e5iT - 6.97e9T^{2} \)
19 \( 1 - 2.19e3T + 1.69e10T^{2} \)
23 \( 1 - 4.99e5iT - 7.83e10T^{2} \)
29 \( 1 + 4.71e5iT - 5.00e11T^{2} \)
31 \( 1 + 1.20e6T + 8.52e11T^{2} \)
37 \( 1 + 9.05e4T + 3.51e12T^{2} \)
41 \( 1 + 5.01e6iT - 7.98e12T^{2} \)
43 \( 1 - 1.25e6T + 1.16e13T^{2} \)
47 \( 1 - 4.26e6iT - 2.38e13T^{2} \)
53 \( 1 + 2.44e6iT - 6.22e13T^{2} \)
59 \( 1 - 3.63e5iT - 1.46e14T^{2} \)
61 \( 1 - 2.52e7T + 1.91e14T^{2} \)
67 \( 1 - 2.86e7T + 4.06e14T^{2} \)
71 \( 1 - 6.10e6iT - 6.45e14T^{2} \)
73 \( 1 - 4.56e7T + 8.06e14T^{2} \)
79 \( 1 + 6.35e6T + 1.51e15T^{2} \)
83 \( 1 - 4.96e7iT - 2.25e15T^{2} \)
89 \( 1 + 6.47e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.05e8T + 7.83e15T^{2} \)
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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

−9.642439115835983813527928931499, −9.129587600079645385863283933492, −8.085770249641542958165560840077, −7.23966548944690581783587801605, −5.40379355756178256480231919083, −5.20267720689926084352089614164, −3.79342077490434733317264271065, −2.35129462199676547056356385662, −0.940079226487895690371920762823, −0.17491507130467527216832065476, 1.93925975751902134222226731424, 2.59711563311538070964753679107, 4.05531241711048958968689057058, 5.09450715579257610846750241607, 6.63138660702401602405265711805, 7.14376729188096505040618348231, 8.105785640978262691118439413512, 9.659221876159246372944660262921, 10.28482694919429912506257048224, 11.02507083037095731552671800850

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