Properties

Label 2-252-7.4-c3-0-4
Degree $2$
Conductor $252$
Sign $0.497 - 0.867i$
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  + (6.22 + 10.7i)5-s + (15.3 − 10.2i)7-s + (−25.5 + 44.2i)11-s + 37.2·13-s + (11.1 − 19.2i)17-s + (−27.1 − 47.0i)19-s + (88.4 + 153. i)23-s + (−14.9 + 25.9i)25-s − 61.0·29-s + (−159. + 277. i)31-s + (206. + 101. i)35-s + (157. + 272. i)37-s + 206.·41-s + 339.·43-s + (71.0 + 123. i)47-s + ⋯
L(s)  = 1  + (0.556 + 0.964i)5-s + (0.831 − 0.556i)7-s + (−0.700 + 1.21i)11-s + 0.794·13-s + (0.158 − 0.274i)17-s + (−0.328 − 0.568i)19-s + (0.801 + 1.38i)23-s + (−0.119 + 0.207i)25-s − 0.391·29-s + (−0.926 + 1.60i)31-s + (0.998 + 0.491i)35-s + (0.699 + 1.21i)37-s + 0.784·41-s + 1.20·43-s + (0.220 + 0.381i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.497 - 0.867i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.497 - 0.867i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.073199053\)
\(L(\frac12)\) \(\approx\) \(2.073199053\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 + (-15.3 + 10.2i)T \)
good5 \( 1 + (-6.22 - 10.7i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (25.5 - 44.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 37.2T + 2.19e3T^{2} \)
17 \( 1 + (-11.1 + 19.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (27.1 + 47.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-88.4 - 153. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 61.0T + 2.43e4T^{2} \)
31 \( 1 + (159. - 277. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-157. - 272. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 206.T + 6.89e4T^{2} \)
43 \( 1 - 339.T + 7.95e4T^{2} \)
47 \( 1 + (-71.0 - 123. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-155. + 268. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (140. - 243. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-271. - 470. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-239. + 415. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 1.10e3T + 3.57e5T^{2} \)
73 \( 1 + (119. - 207. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (580. + 1.00e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 2.93T + 5.71e5T^{2} \)
89 \( 1 + (639. + 1.10e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 79.0T + 9.12e5T^{2} \)
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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

−11.49149772805416660863330565118, −10.76495673866963802751446579221, −10.06914394976891872860408013483, −8.922550816871734110698593545776, −7.54505550077357687680409810046, −6.98974747886539780599569687223, −5.61497349130017592233574578158, −4.47693625995734026485981832455, −2.94317872453654372056680375886, −1.56766565591330115095556364780, 0.899653352061410444339938121024, 2.36621514303353230891687028253, 4.12986023873278750563991157004, 5.46756351735909477918216611409, 5.95478211976424277168533670507, 7.79280165363481422805380170395, 8.612848119817322857133513858412, 9.225651773156585564693379815069, 10.71612868531733593942031781341, 11.26786206274186825970391288918

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