Properties

Label 2-252-7.2-c3-0-4
Degree $2$
Conductor $252$
Sign $0.915 + 0.401i$
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  + (−18.5 + 0.866i)7-s + 89·13-s + (81.5 − 141. i)19-s + (62.5 + 108. i)25-s + (9.5 + 16.4i)31-s + (216.5 − 374. i)37-s + 449·43-s + (341.5 − 32.0i)49-s + (−91 + 157. i)61-s + (−503.5 − 872. i)67-s + (459.5 + 795. i)73-s + (−251.5 + 435. i)79-s + (−1.64e3 + 77.0i)91-s − 1.33e3·97-s + (9.5 − 16.4i)103-s + ⋯
L(s)  = 1  + (−0.998 + 0.0467i)7-s + 1.89·13-s + (0.984 − 1.70i)19-s + (0.5 + 0.866i)25-s + (0.0550 + 0.0953i)31-s + (0.961 − 1.66i)37-s + 1.59·43-s + (0.995 − 0.0934i)49-s + (−0.191 + 0.330i)61-s + (−0.918 − 1.59i)67-s + (0.736 + 1.27i)73-s + (−0.358 + 0.620i)79-s + (−1.89 + 0.0887i)91-s − 1.39·97-s + (0.00908 − 0.0157i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.401i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.915 + 0.401i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.915 + 0.401i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.702673961\)
\(L(\frac12)\) \(\approx\) \(1.702673961\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.5 - 0.866i)T \)
good5 \( 1 + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 89T + 2.19e3T^{2} \)
17 \( 1 + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-81.5 + 141. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 2.43e4T^{2} \)
31 \( 1 + (-9.5 - 16.4i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-216.5 + 374. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 - 449T + 7.95e4T^{2} \)
47 \( 1 + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (91 - 157. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (503.5 + 872. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 + (-459.5 - 795. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (251.5 - 435. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.33e3T + 9.12e5T^{2} \)
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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

−11.34378914664148130294773976324, −10.76707851053638031530109253656, −9.387745442176583721619123490349, −8.901575701749760013159730014988, −7.47715849709593328140660720171, −6.48122715099855660866464534333, −5.51690785068043219668600762510, −3.95433941747557553840356413728, −2.86341616661481952328288138261, −0.874151340891841604116570156248, 1.14117738000301919970041639552, 3.08530979194786457608277993893, 4.07502494686584631679024140890, 5.80619511458025153892959648891, 6.43275887400981311265324539650, 7.80222347106362064644519515669, 8.763820095806108690421341583587, 9.817189925845689694851714872995, 10.61359898254344370671068778424, 11.70590681576842493966553960138

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