Properties

Label 2-252-3.2-c8-0-9
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  + 221. i·5-s − 907.·7-s + 1.94e4i·11-s − 1.91e4·13-s − 1.34e5i·17-s − 1.94e5·19-s + 4.26e4i·23-s + 3.41e5·25-s + 8.14e5i·29-s − 2.03e5·31-s − 2.01e5i·35-s + 1.43e6·37-s − 2.77e6i·41-s − 5.04e6·43-s − 5.42e6i·47-s + ⋯
L(s)  = 1  + 0.354i·5-s − 0.377·7-s + 1.33i·11-s − 0.670·13-s − 1.61i·17-s − 1.49·19-s + 0.152i·23-s + 0.874·25-s + 1.15i·29-s − 0.220·31-s − 0.134i·35-s + 0.767·37-s − 0.982i·41-s − 1.47·43-s − 1.11i·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

\(L(\frac{9}{2})\) \(\approx\) \(1.164226721\)
\(L(\frac12)\) \(\approx\) \(1.164226721\)
\(L(5)\) 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 \( 1 - 221. iT - 3.90e5T^{2} \)
11 \( 1 - 1.94e4iT - 2.14e8T^{2} \)
13 \( 1 + 1.91e4T + 8.15e8T^{2} \)
17 \( 1 + 1.34e5iT - 6.97e9T^{2} \)
19 \( 1 + 1.94e5T + 1.69e10T^{2} \)
23 \( 1 - 4.26e4iT - 7.83e10T^{2} \)
29 \( 1 - 8.14e5iT - 5.00e11T^{2} \)
31 \( 1 + 2.03e5T + 8.52e11T^{2} \)
37 \( 1 - 1.43e6T + 3.51e12T^{2} \)
41 \( 1 + 2.77e6iT - 7.98e12T^{2} \)
43 \( 1 + 5.04e6T + 1.16e13T^{2} \)
47 \( 1 + 5.42e6iT - 2.38e13T^{2} \)
53 \( 1 - 1.34e7iT - 6.22e13T^{2} \)
59 \( 1 + 1.38e7iT - 1.46e14T^{2} \)
61 \( 1 - 2.26e7T + 1.91e14T^{2} \)
67 \( 1 - 3.91e7T + 4.06e14T^{2} \)
71 \( 1 - 3.02e7iT - 6.45e14T^{2} \)
73 \( 1 + 5.42e7T + 8.06e14T^{2} \)
79 \( 1 + 2.10e7T + 1.51e15T^{2} \)
83 \( 1 + 1.30e7iT - 2.25e15T^{2} \)
89 \( 1 + 1.78e7iT - 3.93e15T^{2} \)
97 \( 1 - 6.89e7T + 7.83e15T^{2} \)
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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

−10.33391383868484234973006511339, −9.634985095532114212468963807254, −8.625880128757723326572145219425, −7.16698916871706458926257136759, −6.85327619140510110511055106532, −5.27748473128938841543435203338, −4.36102005646089936906072491926, −2.93974471251272213645087752246, −1.97574438450403793055656299897, −0.32248595840245259041029435080, 0.791819787607958285428916898277, 2.20496528375794865316479348607, 3.48591491073338323797553644457, 4.56284339602248043655386815970, 5.89050596380712173952628689250, 6.58891609525459027356624057944, 8.134416689501160290651561396128, 8.632714502852589963653570805360, 9.861917799796366674053787550501, 10.73485417944268582190554060694

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