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Properties

Label 2-252-63.11-c2-0-5
Degree $2$
Conductor $252$
Sign $0.911 - 0.411i$
Analytic cond. $6.86650$
Root an. cond. $2.62040$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.68 + 2.48i)3-s + (−5.63 − 3.25i)5-s + (6.72 − 1.94i)7-s + (−3.33 − 8.35i)9-s + (8.55 − 4.94i)11-s + (10.3 + 17.8i)13-s + (17.5 − 8.51i)15-s + (7.92 + 4.57i)17-s + (9.38 + 16.2i)19-s + (−6.47 + 19.9i)21-s + (−18.9 − 10.9i)23-s + (8.63 + 14.9i)25-s + (26.3 + 5.76i)27-s + (41.3 + 23.8i)29-s + 27.3·31-s + ⋯
L(s)  = 1  + (−0.560 + 0.827i)3-s + (−1.12 − 0.650i)5-s + (0.960 − 0.277i)7-s + (−0.371 − 0.928i)9-s + (0.778 − 0.449i)11-s + (0.792 + 1.37i)13-s + (1.16 − 0.567i)15-s + (0.466 + 0.269i)17-s + (0.493 + 0.855i)19-s + (−0.308 + 0.951i)21-s + (−0.824 − 0.475i)23-s + (0.345 + 0.598i)25-s + (0.976 + 0.213i)27-s + (1.42 + 0.823i)29-s + 0.882·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.911 - 0.411i$
Analytic conductor: \(6.86650\)
Root analytic conductor: \(2.62040\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1),\ 0.911 - 0.411i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.22101 + 0.262765i\)
\(L(\frac12)\) \(\approx\) \(1.22101 + 0.262765i\)
\(L(2)\) 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 \( 1 + (1.68 - 2.48i)T \)
7 \( 1 + (-6.72 + 1.94i)T \)
good5 \( 1 + (5.63 + 3.25i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-8.55 + 4.94i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-10.3 - 17.8i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (-7.92 - 4.57i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-9.38 - 16.2i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (18.9 + 10.9i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-41.3 - 23.8i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 - 27.3T + 961T^{2} \)
37 \( 1 + (19.5 + 33.8i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-41.0 + 23.6i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-1.87 + 3.24i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + 43.4iT - 2.20e3T^{2} \)
53 \( 1 + (-18.4 - 10.6i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 - 2.89iT - 3.48e3T^{2} \)
61 \( 1 - 110.T + 3.72e3T^{2} \)
67 \( 1 - 46.4T + 4.48e3T^{2} \)
71 \( 1 + 15.9iT - 5.04e3T^{2} \)
73 \( 1 + (65.8 - 114. i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 45.8T + 6.24e3T^{2} \)
83 \( 1 + (100. + 58.2i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (127. - 73.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (20.5 - 35.6i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.80979861224633687747546264130, −11.14263874029503493142701001089, −10.10789646451898342712702089179, −8.776041516453543699963131569966, −8.297389594067880055325280850226, −6.83238157300535751997850467174, −5.56006753283600934250273853424, −4.25793282630620711684482030583, −3.89781243267365103956027297081, −1.08011782731908164451124309278, 1.01860071654275795152020817731, 2.89661639606262744151201733557, 4.46120298306436435236628934932, 5.73302137440999400143815830917, 6.87669974877682513663874046875, 7.83926070989808575850823785160, 8.336362419994330565552519016513, 10.10850736470470025325805972672, 11.23435086966786384691556267388, 11.64058963333056803159820704091

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