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Properties

Label 2-252-63.59-c3-0-20
Degree $2$
Conductor $252$
Sign $-0.284 + 0.958i$
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  + (−0.596 − 5.16i)3-s + 18.4·5-s + (−4.17 − 18.0i)7-s + (−26.2 + 6.15i)9-s − 50.5i·11-s + (68.1 + 39.3i)13-s + (−11.0 − 95.3i)15-s + (−2.61 + 4.52i)17-s + (−23.3 + 13.5i)19-s + (−90.6 + 32.3i)21-s − 58.2i·23-s + 216.·25-s + (47.4 + 132. i)27-s + (−133. + 76.9i)29-s + (−1.78 + 1.03i)31-s + ⋯
L(s)  = 1  + (−0.114 − 0.993i)3-s + 1.65·5-s + (−0.225 − 0.974i)7-s + (−0.973 + 0.227i)9-s − 1.38i·11-s + (1.45 + 0.839i)13-s + (−0.189 − 1.64i)15-s + (−0.0372 + 0.0645i)17-s + (−0.282 + 0.163i)19-s + (−0.941 + 0.335i)21-s − 0.527i·23-s + 1.73·25-s + (0.338 + 0.941i)27-s + (−0.853 + 0.492i)29-s + (−0.0103 + 0.00596i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.284 + 0.958i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.284 + 0.958i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.284 + 0.958i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.157594815\)
\(L(\frac12)\) \(\approx\) \(2.157594815\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.596 + 5.16i)T \)
7 \( 1 + (4.17 + 18.0i)T \)
good5 \( 1 - 18.4T + 125T^{2} \)
11 \( 1 + 50.5iT - 1.33e3T^{2} \)
13 \( 1 + (-68.1 - 39.3i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (2.61 - 4.52i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (23.3 - 13.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 58.2iT - 1.21e4T^{2} \)
29 \( 1 + (133. - 76.9i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (1.78 - 1.03i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (184. + 319. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-199. + 345. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (152. + 263. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (147. - 255. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-431. - 249. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-78.9 - 136. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (628. + 363. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-82.2 - 142. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 157. iT - 3.57e5T^{2} \)
73 \( 1 + (-908. - 524. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (99.3 - 172. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-492. - 853. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (45.0 + 78.0i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-785. + 453. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.02543080760954848075082644648, −10.68320178827736643150803314852, −9.207355635461876007036703948433, −8.521039744436306913284685873464, −7.05898697134968519066906266269, −6.22141803181239223793198761868, −5.61099170304459449423936071621, −3.61562930473202841989976559244, −2.01969117019063048100524735361, −0.898211415882512556603282008343, 1.86828930504038538068438558933, 3.16423062342732437615902364531, 4.84169347522354041393044961213, 5.72829176658262015237532722482, 6.42576190299005994969458438817, 8.334932178202821078111216408171, 9.335917424883864797289596153236, 9.827502702106417610137501832093, 10.66212283097959380230456636048, 11.77918356372328895078582717779

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