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Properties

Label 2-252-63.58-c3-0-18
Degree $2$
Conductor $252$
Sign $0.279 + 0.960i$
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  + (3.26 − 4.03i)3-s + (−0.275 − 0.476i)5-s + (18.2 + 3.03i)7-s + (−5.63 − 26.4i)9-s + (3.66 − 6.34i)11-s + (16.5 − 28.5i)13-s + (−2.82 − 0.446i)15-s + (17.5 + 30.3i)17-s + (17.5 − 30.4i)19-s + (71.9 − 63.8i)21-s + (36.8 + 63.8i)23-s + (62.3 − 107. i)25-s + (−125. − 63.5i)27-s + (−108. − 187. i)29-s + 69.4·31-s + ⋯
L(s)  = 1  + (0.629 − 0.777i)3-s + (−0.0245 − 0.0426i)5-s + (0.986 + 0.163i)7-s + (−0.208 − 0.978i)9-s + (0.100 − 0.173i)11-s + (0.352 − 0.609i)13-s + (−0.0485 − 0.00767i)15-s + (0.250 + 0.433i)17-s + (0.212 − 0.367i)19-s + (0.748 − 0.663i)21-s + (0.334 + 0.578i)23-s + (0.498 − 0.863i)25-s + (−0.891 − 0.453i)27-s + (−0.692 − 1.19i)29-s + 0.402·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.279 + 0.960i$
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: \((2,\ 252,\ (\ :3/2),\ 0.279 + 0.960i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.433707743\)
\(L(\frac12)\) \(\approx\) \(2.433707743\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.26 + 4.03i)T \)
7 \( 1 + (-18.2 - 3.03i)T \)
good5 \( 1 + (0.275 + 0.476i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-3.66 + 6.34i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-16.5 + 28.5i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-17.5 - 30.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-17.5 + 30.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-36.8 - 63.8i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (108. + 187. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 69.4T + 2.97e4T^{2} \)
37 \( 1 + (72.3 - 125. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-90.0 + 156. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (194. + 336. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 222.T + 1.03e5T^{2} \)
53 \( 1 + (-227. - 393. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + 352.T + 2.05e5T^{2} \)
61 \( 1 - 227.T + 2.26e5T^{2} \)
67 \( 1 - 304.T + 3.00e5T^{2} \)
71 \( 1 - 282.T + 3.57e5T^{2} \)
73 \( 1 + (94.2 + 163. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 - 1.12e3T + 4.93e5T^{2} \)
83 \( 1 + (-268. - 465. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (691. - 1.19e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-113. - 196. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.58757598889707442491524863710, −10.52439985372301228425714747392, −9.223978345234135761147143573576, −8.301662937629844857279969402980, −7.68834620439424743671146754289, −6.46416738921285380440411402748, −5.28865104659970219754416279894, −3.75638129447822652534733478658, −2.35304510761571149892531655731, −0.998758314816105237057293795179, 1.67135483526330179261335437152, 3.24030524733946704027039477700, 4.45627280754870433033382911197, 5.33050469832411697605114222031, 7.00512877285326475084985647001, 8.057031672264220466843189811402, 8.907189267600796877336393223896, 9.806416016245542903377128807869, 10.90052020917306432617557046430, 11.47414154766484441747943100360

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