Properties

Label 2-252-63.47-c3-0-19
Degree $2$
Conductor $252$
Sign $0.993 - 0.116i$
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  + (4.25 + 2.98i)3-s + 13.0·5-s + (15.3 − 10.3i)7-s + (9.23 + 25.3i)9-s − 52.7i·11-s + (31.8 − 18.3i)13-s + (55.7 + 39.0i)15-s + (20.4 + 35.4i)17-s + (−108. − 62.8i)19-s + (96.2 + 1.79i)21-s − 2.29i·23-s + 46.2·25-s + (−36.3 + 135. i)27-s + (−151. − 87.5i)29-s + (232. + 134. i)31-s + ⋯
L(s)  = 1  + (0.819 + 0.573i)3-s + 1.17·5-s + (0.829 − 0.558i)7-s + (0.342 + 0.939i)9-s − 1.44i·11-s + (0.679 − 0.392i)13-s + (0.958 + 0.671i)15-s + (0.292 + 0.506i)17-s + (−1.31 − 0.758i)19-s + (0.999 + 0.0186i)21-s − 0.0208i·23-s + 0.370·25-s + (−0.258 + 0.965i)27-s + (−0.971 − 0.560i)29-s + (1.34 + 0.778i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.993 - 0.116i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.993 - 0.116i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.156825454\)
\(L(\frac12)\) \(\approx\) \(3.156825454\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.25 - 2.98i)T \)
7 \( 1 + (-15.3 + 10.3i)T \)
good5 \( 1 - 13.0T + 125T^{2} \)
11 \( 1 + 52.7iT - 1.33e3T^{2} \)
13 \( 1 + (-31.8 + 18.3i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-20.4 - 35.4i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (108. + 62.8i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 2.29iT - 1.21e4T^{2} \)
29 \( 1 + (151. + 87.5i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-232. - 134. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (191. - 331. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-75.5 - 130. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-23.2 + 40.2i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-186. - 322. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (539. - 311. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-2.00 + 3.47i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-675. + 389. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-219. + 380. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 800. iT - 3.57e5T^{2} \)
73 \( 1 + (428. - 247. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (70.1 + 121. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (506. - 877. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + (211. - 366. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-264. - 152. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.17070268310402891194516852998, −10.64419321807773618550681135826, −9.731129869048124740651746645571, −8.573740433183627356955624210314, −8.135789761078044453562651857385, −6.47368252367642062287782970628, −5.40004281156923098626818023934, −4.15654580302303021425690421942, −2.84366081968683564312917111588, −1.42049933400569316896794852911, 1.71400414228274640472637053390, 2.25348867577478584667842647948, 4.13097054905534970344041139031, 5.54846330458529996267214229564, 6.61905607179588987768980367832, 7.69853433836969170241186550426, 8.749690804682203708230341253252, 9.487762251858200421807687826115, 10.42645183861983948772714627608, 11.81443363192411728054332370861

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