↑

Properties

Label 2-252-63.59-c3-0-19
Degree $2$
Conductor $252$
Sign $-0.462 + 0.886i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.17 + 5.06i)3-s + 10.3·5-s + (−12.2 − 13.8i)7-s + (−24.2 − 11.9i)9-s + 3.35i·11-s + (−73.0 − 42.1i)13-s + (−12.1 + 52.1i)15-s + (−37.1 + 64.3i)17-s + (7.67 − 4.43i)19-s + (84.7 − 45.6i)21-s − 137. i·23-s − 18.6·25-s + (88.7 − 108. i)27-s + (−127. + 73.8i)29-s + (32.8 − 18.9i)31-s + ⋯
L(s)  = 1  + (−0.226 + 0.974i)3-s + 0.922·5-s + (−0.660 − 0.750i)7-s + (−0.897 − 0.440i)9-s + 0.0919i·11-s + (−1.55 − 0.900i)13-s + (−0.208 + 0.898i)15-s + (−0.530 + 0.918i)17-s + (0.0926 − 0.0535i)19-s + (0.880 − 0.473i)21-s − 1.25i·23-s − 0.149·25-s + (0.632 − 0.774i)27-s + (−0.818 + 0.472i)29-s + (0.190 − 0.109i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.462 + 0.886i$
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: \((2,\ 252,\ (\ :3/2),\ -0.462 + 0.886i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4487634660\)
\(L(\frac12)\) \(\approx\) \(0.4487634660\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.17 - 5.06i)T \)
7 \( 1 + (12.2 + 13.8i)T \)
good5 \( 1 - 10.3T + 125T^{2} \)
11 \( 1 - 3.35iT - 1.33e3T^{2} \)
13 \( 1 + (73.0 + 42.1i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (37.1 - 64.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-7.67 + 4.43i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + 137. iT - 1.21e4T^{2} \)
29 \( 1 + (127. - 73.8i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-32.8 + 18.9i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (35.7 + 61.9i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-98.5 + 170. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (102. + 177. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-181. + 314. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (586. + 338. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-282. - 488. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (164. + 95.2i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-428. - 741. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 53.7iT - 3.57e5T^{2} \)
73 \( 1 + (-234. - 135. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (260. - 451. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (66.9 + 115. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (417. + 722. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (591. - 341. i)T + (4.56e5 - 7.90e5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.92217455132969130160622090488, −10.15575580690512194569311341545, −9.751142592884258733644353587095, −8.611180501563879696007886470176, −7.16232762217146320635989115267, −6.03679888091881079575286450458, −5.06340517393839435426847513699, −3.86755820764555168391297229992, −2.48106562403381842182264153537, −0.16548963919426808330305421408, 1.85902216358287802454027467119, 2.79301591976078432392652737182, 4.99248972107723319042876718072, 5.96502194535107058848936314894, 6.82105378611107734113099936365, 7.79949965195989082458319572072, 9.383707249282964000831606647880, 9.562836730952091760974732629946, 11.25216091059911046683885303792, 11.99901577671750799318297369926

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