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Properties

Label 2-252-28.27-c3-0-15
Degree $2$
Conductor $252$
Sign $0.831 - 0.554i$
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  + (−2.82 − 0.0488i)2-s + (7.99 + 0.276i)4-s − 16.6i·5-s + (−15.0 + 10.8i)7-s + (−22.5 − 1.17i)8-s + (−0.813 + 47.0i)10-s + 64.0i·11-s − 28.6i·13-s + (43.0 − 29.8i)14-s + (63.8 + 4.41i)16-s + 82.9i·17-s + 17.1·19-s + (4.60 − 133. i)20-s + (3.12 − 181. i)22-s − 95.0i·23-s + ⋯
L(s)  = 1  + (−0.999 − 0.0172i)2-s + (0.999 + 0.0345i)4-s − 1.48i·5-s + (−0.812 + 0.583i)7-s + (−0.998 − 0.0517i)8-s + (−0.0257 + 1.48i)10-s + 1.75i·11-s − 0.612i·13-s + (0.822 − 0.569i)14-s + (0.997 + 0.0690i)16-s + 1.18i·17-s + 0.206·19-s + (0.0514 − 1.48i)20-s + (0.0303 − 1.75i)22-s − 0.861i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.831 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.831 - 0.554i$
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: \((2,\ 252,\ (\ :3/2),\ 0.831 - 0.554i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8761498382\)
\(L(\frac12)\) \(\approx\) \(0.8761498382\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.82 + 0.0488i)T \)
3 \( 1 \)
7 \( 1 + (15.0 - 10.8i)T \)
good5 \( 1 + 16.6iT - 125T^{2} \)
11 \( 1 - 64.0iT - 1.33e3T^{2} \)
13 \( 1 + 28.6iT - 2.19e3T^{2} \)
17 \( 1 - 82.9iT - 4.91e3T^{2} \)
19 \( 1 - 17.1T + 6.85e3T^{2} \)
23 \( 1 + 95.0iT - 1.21e4T^{2} \)
29 \( 1 - 197.T + 2.43e4T^{2} \)
31 \( 1 - 153.T + 2.97e4T^{2} \)
37 \( 1 - 10.7T + 5.06e4T^{2} \)
41 \( 1 - 41.1iT - 6.89e4T^{2} \)
43 \( 1 - 412. iT - 7.95e4T^{2} \)
47 \( 1 - 477.T + 1.03e5T^{2} \)
53 \( 1 - 35.2T + 1.48e5T^{2} \)
59 \( 1 - 494.T + 2.05e5T^{2} \)
61 \( 1 - 294. iT - 2.26e5T^{2} \)
67 \( 1 - 207. iT - 3.00e5T^{2} \)
71 \( 1 - 534. iT - 3.57e5T^{2} \)
73 \( 1 - 582. iT - 3.89e5T^{2} \)
79 \( 1 - 311. iT - 4.93e5T^{2} \)
83 \( 1 + 1.31e3T + 5.71e5T^{2} \)
89 \( 1 + 616. iT - 7.04e5T^{2} \)
97 \( 1 + 104. iT - 9.12e5T^{2} \)
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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

−11.89785096180798736231953067367, −10.27429715323994176707744700745, −9.757067401563658347618142614603, −8.764789622886304327611150177578, −8.124190506737474257960454789702, −6.83137161046901404995930990148, −5.71340585969475609092237494546, −4.40159006538679073359903616905, −2.49881523659573326761637278378, −1.06845249568035848096738501637, 0.58288508763147036600146034727, 2.72734083204300439908986842426, 3.47414121977849599549536543450, 5.90626111642497225702775962526, 6.74765327381329806995006064506, 7.40551157133223079337663792435, 8.677080852217374791818948738325, 9.717608921464252938358689671041, 10.51932567919933633004498338513, 11.21838272472929145563926300869

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