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Properties

Label 2-252-252.131-c2-0-13
Degree $2$
Conductor $252$
Sign $0.998 - 0.0548i$
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  + (−1.47 − 1.35i)2-s + (−1.75 − 2.43i)3-s + (0.339 + 3.98i)4-s + (0.918 + 1.59i)5-s + (−0.715 + 5.95i)6-s + (−4.87 − 5.02i)7-s + (4.89 − 6.33i)8-s + (−2.86 + 8.53i)9-s + (0.799 − 3.58i)10-s + (−6.17 + 10.6i)11-s + (9.11 − 7.80i)12-s + (7.78 + 4.49i)13-s + (0.378 + 13.9i)14-s + (2.26 − 5.02i)15-s + (−15.7 + 2.70i)16-s + (8.35 + 14.4i)17-s + ⋯
L(s)  = 1  + (−0.736 − 0.676i)2-s + (−0.583 − 0.811i)3-s + (0.0849 + 0.996i)4-s + (0.183 + 0.318i)5-s + (−0.119 + 0.992i)6-s + (−0.696 − 0.717i)7-s + (0.611 − 0.791i)8-s + (−0.318 + 0.947i)9-s + (0.0799 − 0.358i)10-s + (−0.561 + 0.972i)11-s + (0.759 − 0.650i)12-s + (0.598 + 0.345i)13-s + (0.0270 + 0.999i)14-s + (0.151 − 0.334i)15-s + (−0.985 + 0.169i)16-s + (0.491 + 0.851i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0548i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.998 - 0.0548i$
Analytic conductor: \(6.86650\)
Root analytic conductor: \(2.62040\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1),\ 0.998 - 0.0548i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.678550 + 0.0186358i\)
\(L(\frac12)\) \(\approx\) \(0.678550 + 0.0186358i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.47 + 1.35i)T \)
3 \( 1 + (1.75 + 2.43i)T \)
7 \( 1 + (4.87 + 5.02i)T \)
good5 \( 1 + (-0.918 - 1.59i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (6.17 - 10.6i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-7.78 - 4.49i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-8.35 - 14.4i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-0.794 + 1.37i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-10.1 - 17.5i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (7.64 - 4.41i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 18.4T + 961T^{2} \)
37 \( 1 + (-5.48 + 9.49i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-10.6 + 18.3i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-59.4 + 34.3i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 - 74.3iT - 2.20e3T^{2} \)
53 \( 1 + (34.0 - 19.6i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 - 88.1iT - 3.48e3T^{2} \)
61 \( 1 - 49.3iT - 3.72e3T^{2} \)
67 \( 1 - 101. iT - 4.48e3T^{2} \)
71 \( 1 + 7.14T + 5.04e3T^{2} \)
73 \( 1 + (116. - 67.2i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + 62.5iT - 6.24e3T^{2} \)
83 \( 1 + (-77.8 + 44.9i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + (-27.0 + 46.8i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (13.2 - 7.67i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−11.78911334604696517065901428701, −10.66381867361989045085164614487, −10.28178955847884715681219817007, −9.044468665321751481666052580925, −7.71525514703526345477179795320, −7.11331898004319661743462872607, −6.03336044154046359191377712278, −4.23520967960369364885061181826, −2.68117037938836680608345578033, −1.21367171794419085916815832803, 0.56312638737172253272681195952, 3.09145191188479623808520231072, 4.99316280911703881898327040800, 5.74834745696130761150238524183, 6.56653664588820302095494337751, 8.127584904868110235826253356243, 9.045696269913247238570385928236, 9.695626880709185382451137478291, 10.70273754715394662967735791481, 11.44388884063455604289597999630

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