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Properties

Label 2-252-63.41-c3-0-18
Degree $2$
Conductor $252$
Sign $0.864 + 0.502i$
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  + (5.09 + 1.03i)3-s + (10.5 − 18.2i)5-s + (17.6 + 5.46i)7-s + (24.8 + 10.5i)9-s + (−21.5 + 12.4i)11-s + (52.5 + 30.3i)13-s + (72.7 − 82.1i)15-s − 117.·17-s − 104. i·19-s + (84.4 + 46.1i)21-s + (−17.4 − 10.0i)23-s + (−160. − 277. i)25-s + (115. + 79.5i)27-s + (−24.2 + 14.0i)29-s + (216. + 125. i)31-s + ⋯
L(s)  = 1  + (0.979 + 0.199i)3-s + (0.944 − 1.63i)5-s + (0.955 + 0.295i)7-s + (0.920 + 0.390i)9-s + (−0.589 + 0.340i)11-s + (1.12 + 0.647i)13-s + (1.25 − 1.41i)15-s − 1.67·17-s − 1.26i·19-s + (0.877 + 0.479i)21-s + (−0.158 − 0.0915i)23-s + (−1.28 − 2.22i)25-s + (0.823 + 0.566i)27-s + (−0.155 + 0.0897i)29-s + (1.25 + 0.724i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.502i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.864 + 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.864 + 0.502i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.864 + 0.502i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.214431821\)
\(L(\frac12)\) \(\approx\) \(3.214431821\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.09 - 1.03i)T \)
7 \( 1 + (-17.6 - 5.46i)T \)
good5 \( 1 + (-10.5 + 18.2i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (21.5 - 12.4i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-52.5 - 30.3i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 117.T + 4.91e3T^{2} \)
19 \( 1 + 104. iT - 6.85e3T^{2} \)
23 \( 1 + (17.4 + 10.0i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (24.2 - 14.0i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-216. - 125. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 18.2T + 5.06e4T^{2} \)
41 \( 1 + (153. - 265. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-74.5 - 129. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (108. + 188. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 116. iT - 1.48e5T^{2} \)
59 \( 1 + (-38.3 + 66.4i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (493. - 285. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (33.8 - 58.6i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 796. iT - 3.57e5T^{2} \)
73 \( 1 + 710. iT - 3.89e5T^{2} \)
79 \( 1 + (40.0 + 69.3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-57.6 - 99.8i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.05e3T + 7.04e5T^{2} \)
97 \( 1 + (444. - 256. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.54708124662273919696901110360, −10.39314754834309836088635172813, −9.107984406358831948570079695055, −8.859907094748764673554587349779, −8.026592790321649189710714916390, −6.43617643348297971536847916058, −4.88613316429625461253947997008, −4.50008447968699557862323478628, −2.35058942571740308435911165603, −1.39101417777749313869619982597, 1.76709876678983432302986784801, 2.76774584734251695093653498888, 3.96390984072608706102240228373, 5.80246076256315133411277205585, 6.75291407407646295584380378782, 7.80110102645555650171750034597, 8.609200894629178622830232511531, 9.954901549566884316554114062071, 10.62694983040377843572164506194, 11.34542561123950358687794842453

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