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Properties

Label 2-252-28.27-c3-0-20
Degree $2$
Conductor $252$
Sign $0.719 - 0.694i$
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  + (−2.16 + 1.81i)2-s + (1.38 − 7.87i)4-s + 4.47i·5-s + (−14.9 − 10.8i)7-s + (11.3 + 19.5i)8-s + (−8.13 − 9.69i)10-s + 7.61i·11-s − 13.3i·13-s + (52.2 − 3.65i)14-s + (−60.1 − 21.8i)16-s + 55.3i·17-s + 73.9·19-s + (35.2 + 6.20i)20-s + (−13.8 − 16.4i)22-s − 133. i·23-s + ⋯
L(s)  = 1  + (−0.765 + 0.642i)2-s + (0.173 − 0.984i)4-s + 0.400i·5-s + (−0.808 − 0.587i)7-s + (0.500 + 0.865i)8-s + (−0.257 − 0.306i)10-s + 0.208i·11-s − 0.285i·13-s + (0.997 − 0.0698i)14-s + (−0.939 − 0.341i)16-s + 0.789i·17-s + 0.892·19-s + (0.394 + 0.0693i)20-s + (−0.134 − 0.159i)22-s − 1.20i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.719 - 0.694i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.719 - 0.694i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.062256780\)
\(L(\frac12)\) \(\approx\) \(1.062256780\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.16 - 1.81i)T \)
3 \( 1 \)
7 \( 1 + (14.9 + 10.8i)T \)
good5 \( 1 - 4.47iT - 125T^{2} \)
11 \( 1 - 7.61iT - 1.33e3T^{2} \)
13 \( 1 + 13.3iT - 2.19e3T^{2} \)
17 \( 1 - 55.3iT - 4.91e3T^{2} \)
19 \( 1 - 73.9T + 6.85e3T^{2} \)
23 \( 1 + 133. iT - 1.21e4T^{2} \)
29 \( 1 + 23.3T + 2.43e4T^{2} \)
31 \( 1 - 241.T + 2.97e4T^{2} \)
37 \( 1 - 178.T + 5.06e4T^{2} \)
41 \( 1 - 494. iT - 6.89e4T^{2} \)
43 \( 1 - 72.6iT - 7.95e4T^{2} \)
47 \( 1 - 59.7T + 1.03e5T^{2} \)
53 \( 1 - 569.T + 1.48e5T^{2} \)
59 \( 1 - 59.5T + 2.05e5T^{2} \)
61 \( 1 - 629. iT - 2.26e5T^{2} \)
67 \( 1 + 599. iT - 3.00e5T^{2} \)
71 \( 1 - 407. iT - 3.57e5T^{2} \)
73 \( 1 - 680. iT - 3.89e5T^{2} \)
79 \( 1 + 1.08e3iT - 4.93e5T^{2} \)
83 \( 1 - 935.T + 5.71e5T^{2} \)
89 \( 1 + 12.1iT - 7.04e5T^{2} \)
97 \( 1 + 1.43e3iT - 9.12e5T^{2} \)
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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

−11.43536661076889744325621122019, −10.36312617357827543959207495156, −9.934046027722839634989662038299, −8.748827043562978758312905934168, −7.72739401948624476022546054394, −6.77307886155250361667842023047, −6.02778726709213102593630432760, −4.52912191135817382355309750970, −2.83647061761157538630094380264, −0.891186900787480453931902936353, 0.800830088635151396195879915041, 2.51071359639766773604571765444, 3.65389190765777015438447931782, 5.24908457385107474038580416150, 6.67938535791850762924178408316, 7.72593702243965124878516232107, 8.936960658838878861865395495832, 9.425379890670869823728314071231, 10.39461818865324993323313865870, 11.63027085429766176871924202203

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