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Properties

Label 2-252-84.23-c1-0-2
Degree $2$
Conductor $252$
Sign $0.983 + 0.180i$
Analytic cond. $2.01223$
Root an. cond. $1.41853$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.857 − 1.12i)2-s + (−0.530 + 1.92i)4-s + (−0.604 − 0.349i)5-s + (1.16 + 2.37i)7-s + (2.62 − 1.05i)8-s + (0.125 + 0.979i)10-s + (1.27 + 2.21i)11-s + 1.88·13-s + (1.67 − 3.34i)14-s + (−3.43 − 2.04i)16-s + (3.44 − 1.98i)17-s + (6.11 + 3.52i)19-s + (0.994 − 0.980i)20-s + (1.39 − 3.32i)22-s + (−2.01 + 3.48i)23-s + ⋯
L(s)  = 1  + (−0.606 − 0.795i)2-s + (−0.265 + 0.964i)4-s + (−0.270 − 0.156i)5-s + (0.439 + 0.898i)7-s + (0.927 − 0.373i)8-s + (0.0397 + 0.309i)10-s + (0.384 + 0.666i)11-s + 0.521·13-s + (0.448 − 0.893i)14-s + (−0.859 − 0.511i)16-s + (0.834 − 0.481i)17-s + (1.40 + 0.809i)19-s + (0.222 − 0.219i)20-s + (0.296 − 0.709i)22-s + (−0.419 + 0.727i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.983 + 0.180i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.983 + 0.180i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.943507 - 0.0860364i\)
\(L(\frac12)\) \(\approx\) \(0.943507 - 0.0860364i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.857 + 1.12i)T \)
3 \( 1 \)
7 \( 1 + (-1.16 - 2.37i)T \)
good5 \( 1 + (0.604 + 0.349i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.27 - 2.21i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.88T + 13T^{2} \)
17 \( 1 + (-3.44 + 1.98i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.11 - 3.52i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.01 - 3.48i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.86iT - 29T^{2} \)
31 \( 1 + (-0.815 + 0.470i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.74 + 6.49i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.6iT - 41T^{2} \)
43 \( 1 + 3.97iT - 43T^{2} \)
47 \( 1 + (4.45 - 7.71i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.458 + 0.264i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.65 + 11.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.18 - 8.97i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.35 - 1.36i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.51T + 71T^{2} \)
73 \( 1 + (-1.37 - 2.38i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (11.7 + 6.76i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 17.1T + 83T^{2} \)
89 \( 1 + (10.2 + 5.93i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−11.90331151259065005924672372868, −11.28969263117928607280734042022, −9.899513518110394502080584472319, −9.369988915104405850470978833739, −8.162123058783152843614219223838, −7.54183538576819843170734150886, −5.83686594931863441061942665006, −4.45889888571321694799330175836, −3.12860782209932450207143080456, −1.56981199857865061283647040091, 1.12927318895203837137729564340, 3.63619957167902880557338250339, 5.02060778938155254701134424015, 6.20170330409886904020102271945, 7.28211795969962747577546665272, 8.022733570848361877239153035738, 9.022119801429092544670867209044, 10.11159390979817274411749657423, 10.93786317268426021970380791966, 11.77869504048454445233750947418

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