Properties

Label 2-252-3.2-c8-0-0
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  − 899. i·5-s + 907.·7-s + 2.19e4i·11-s + 2.08e4·13-s + 5.49e4i·17-s − 1.12e5·19-s − 3.64e5i·23-s − 4.18e5·25-s − 4.93e5i·29-s − 9.47e5·31-s − 8.16e5i·35-s − 2.31e6·37-s + 3.36e5i·41-s − 1.64e5·43-s − 1.21e6i·47-s + ⋯
L(s)  = 1  − 1.43i·5-s + 0.377·7-s + 1.49i·11-s + 0.730·13-s + 0.657i·17-s − 0.861·19-s − 1.30i·23-s − 1.07·25-s − 0.697i·29-s − 1.02·31-s − 0.543i·35-s − 1.23·37-s + 0.118i·41-s − 0.0480·43-s − 0.249i·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

\(L(\frac{9}{2})\) \(\approx\) \(0.4133446162\)
\(L(\frac12)\) \(\approx\) \(0.4133446162\)
\(L(5)\) 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 \( 1 + 899. iT - 3.90e5T^{2} \)
11 \( 1 - 2.19e4iT - 2.14e8T^{2} \)
13 \( 1 - 2.08e4T + 8.15e8T^{2} \)
17 \( 1 - 5.49e4iT - 6.97e9T^{2} \)
19 \( 1 + 1.12e5T + 1.69e10T^{2} \)
23 \( 1 + 3.64e5iT - 7.83e10T^{2} \)
29 \( 1 + 4.93e5iT - 5.00e11T^{2} \)
31 \( 1 + 9.47e5T + 8.52e11T^{2} \)
37 \( 1 + 2.31e6T + 3.51e12T^{2} \)
41 \( 1 - 3.36e5iT - 7.98e12T^{2} \)
43 \( 1 + 1.64e5T + 1.16e13T^{2} \)
47 \( 1 + 1.21e6iT - 2.38e13T^{2} \)
53 \( 1 - 2.50e6iT - 6.22e13T^{2} \)
59 \( 1 - 1.27e7iT - 1.46e14T^{2} \)
61 \( 1 + 1.01e7T + 1.91e14T^{2} \)
67 \( 1 - 7.53e6T + 4.06e14T^{2} \)
71 \( 1 - 4.21e7iT - 6.45e14T^{2} \)
73 \( 1 + 3.47e7T + 8.06e14T^{2} \)
79 \( 1 - 1.95e6T + 1.51e15T^{2} \)
83 \( 1 - 2.01e7iT - 2.25e15T^{2} \)
89 \( 1 + 1.15e7iT - 3.93e15T^{2} \)
97 \( 1 + 9.91e7T + 7.83e15T^{2} \)
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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

−10.89308675502760857984914834005, −9.935597379088728372658784411528, −8.825261574478323199231087400196, −8.310307412319677933655806212906, −7.06903138775570110460038190414, −5.79932141718572417681909093396, −4.68948071877759906999074675168, −4.08106782300973884549707072318, −2.11645703437457627308618045026, −1.25597391849202029730914831216, 0.086539206666428555650084594174, 1.63869680012242266053083543562, 3.02307442500337916306354011275, 3.68918042958340598088620173073, 5.39323018405936891593283595256, 6.33290608028587996346745492202, 7.23803722288364701991785634768, 8.297185802010610227474198359098, 9.291040308304766026820165641142, 10.74827532772450260759052783720

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