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Properties

Label 2-252-63.58-c3-0-19
Degree $2$
Conductor $252$
Sign $-0.543 + 0.839i$
Analytic cond. $14.8684$
Root an. cond. $3.85596$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.340 − 5.18i)3-s + (4.29 + 7.44i)5-s + (−11.6 + 14.3i)7-s + (−26.7 − 3.53i)9-s + (16.1 − 28.0i)11-s + (33.6 − 58.2i)13-s + (40.0 − 19.7i)15-s + (−54.8 − 94.9i)17-s + (10.0 − 17.3i)19-s + (70.6 + 65.3i)21-s + (−78.8 − 136. i)23-s + (25.5 − 44.2i)25-s + (−27.4 + 137. i)27-s + (103. + 180. i)29-s − 273.·31-s + ⋯
L(s)  = 1  + (0.0655 − 0.997i)3-s + (0.384 + 0.665i)5-s + (−0.629 + 0.776i)7-s + (−0.991 − 0.130i)9-s + (0.443 − 0.768i)11-s + (0.717 − 1.24i)13-s + (0.689 − 0.339i)15-s + (−0.782 − 1.35i)17-s + (0.121 − 0.209i)19-s + (0.733 + 0.679i)21-s + (−0.715 − 1.23i)23-s + (0.204 − 0.354i)25-s + (−0.195 + 0.980i)27-s + (0.665 + 1.15i)29-s − 1.58·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.543 + 0.839i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.543 + 0.839i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.318330900\)
\(L(\frac12)\) \(\approx\) \(1.318330900\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.340 + 5.18i)T \)
7 \( 1 + (11.6 - 14.3i)T \)
good5 \( 1 + (-4.29 - 7.44i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-16.1 + 28.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-33.6 + 58.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (54.8 + 94.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-10.0 + 17.3i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (78.8 + 136. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-103. - 180. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 273.T + 2.97e4T^{2} \)
37 \( 1 + (-92.7 + 160. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (64.1 - 111. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (135. + 233. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 184.T + 1.03e5T^{2} \)
53 \( 1 + (-92.9 - 160. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 179.T + 2.05e5T^{2} \)
61 \( 1 - 607.T + 2.26e5T^{2} \)
67 \( 1 + 210.T + 3.00e5T^{2} \)
71 \( 1 - 1.09e3T + 3.57e5T^{2} \)
73 \( 1 + (248. + 430. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + 1.30e3T + 4.93e5T^{2} \)
83 \( 1 + (-43.7 - 75.7i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (-283. + 491. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-310. - 538. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.32982655138983681382130914682, −10.48652769999244705677213680558, −9.088363101293748144318085301628, −8.435514562406081827531408553972, −7.06538473090439407105203311811, −6.31097271076426830921561160035, −5.47825555445572862397329497544, −3.25310714230764812674323400575, −2.40883726016726788032812662578, −0.50859074233810181850752967638, 1.70258545086776144217078461691, 3.77332709809358694030394807834, 4.34204558251522767098611168957, 5.76306500847066219255451291070, 6.79007156992649852215766367079, 8.293797576916536708900129517462, 9.319439501042152034607036850730, 9.797447667445654701220214750759, 10.86344827147648991007976053081, 11.77059673437912209880641795971

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