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Properties

Label 2-252-12.11-c3-0-20
Degree $2$
Conductor $252$
Sign $0.849 - 0.528i$
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 + 0.0833i)2-s + (7.98 + 0.471i)4-s + 8.62i·5-s − 7i·7-s + (22.5 + 1.99i)8-s + (−0.719 + 24.3i)10-s + 44.2·11-s − 3.70·13-s + (0.583 − 19.7i)14-s + (63.5 + 7.53i)16-s + 35.8i·17-s + 88.8i·19-s + (−4.06 + 68.8i)20-s + (125. + 3.69i)22-s + 41.2·23-s + ⋯
L(s)  = 1  + (0.999 + 0.0294i)2-s + (0.998 + 0.0589i)4-s + 0.771i·5-s − 0.377i·7-s + (0.996 + 0.0883i)8-s + (−0.0227 + 0.771i)10-s + 1.21·11-s − 0.0790·13-s + (0.0111 − 0.377i)14-s + (0.993 + 0.117i)16-s + 0.511i·17-s + 1.07i·19-s + (−0.0454 + 0.770i)20-s + (1.21 + 0.0357i)22-s + 0.373·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.528i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.849 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.849 - 0.528i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.849 - 0.528i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.684733888\)
\(L(\frac12)\) \(\approx\) \(3.684733888\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.82 - 0.0833i)T \)
3 \( 1 \)
7 \( 1 + 7iT \)
good5 \( 1 - 8.62iT - 125T^{2} \)
11 \( 1 - 44.2T + 1.33e3T^{2} \)
13 \( 1 + 3.70T + 2.19e3T^{2} \)
17 \( 1 - 35.8iT - 4.91e3T^{2} \)
19 \( 1 - 88.8iT - 6.85e3T^{2} \)
23 \( 1 - 41.2T + 1.21e4T^{2} \)
29 \( 1 - 12.7iT - 2.43e4T^{2} \)
31 \( 1 - 67.8iT - 2.97e4T^{2} \)
37 \( 1 + 339.T + 5.06e4T^{2} \)
41 \( 1 + 450. iT - 6.89e4T^{2} \)
43 \( 1 + 60.2iT - 7.95e4T^{2} \)
47 \( 1 - 243.T + 1.03e5T^{2} \)
53 \( 1 + 587. iT - 1.48e5T^{2} \)
59 \( 1 - 393.T + 2.05e5T^{2} \)
61 \( 1 + 890.T + 2.26e5T^{2} \)
67 \( 1 - 497. iT - 3.00e5T^{2} \)
71 \( 1 - 94.3T + 3.57e5T^{2} \)
73 \( 1 + 636.T + 3.89e5T^{2} \)
79 \( 1 + 1.11e3iT - 4.93e5T^{2} \)
83 \( 1 + 1.16e3T + 5.71e5T^{2} \)
89 \( 1 - 328. iT - 7.04e5T^{2} \)
97 \( 1 + 467.T + 9.12e5T^{2} \)
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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

−11.85866540130356563009065054590, −10.82016637698091661209013284897, −10.18390842052902922335760320851, −8.673131398147321743012037710395, −7.27692139345359080761834795796, −6.65150658538958646942149847470, −5.57368784545890510296235150133, −4.12671913034196727552262540118, −3.28418365575743963433900486853, −1.67429779386079897610820344746, 1.27179738463290116088452672320, 2.87431504618195815451693641314, 4.29564479611678306205695022419, 5.13854734831091799991224006194, 6.31397306218871522613210530943, 7.26958083324307888641768967520, 8.672241311642155601495249010802, 9.519617460676090917966973928164, 10.92400943024371615728355427644, 11.78286925169339029293523715743

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