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Properties

Label 2-504-56.19-c1-0-20
Degree $2$
Conductor $504$
Sign $0.351 + 0.936i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.34 + 0.434i)2-s + (1.62 − 1.16i)4-s + (−1.44 + 2.49i)5-s + (−2.63 + 0.194i)7-s + (−1.67 + 2.27i)8-s + (0.856 − 3.98i)10-s + (−2.91 − 5.04i)11-s + 1.04·13-s + (3.46 − 1.40i)14-s + (1.26 − 3.79i)16-s + (5.91 − 3.41i)17-s + (−0.589 − 0.340i)19-s + (0.577 + 5.73i)20-s + (6.11 + 5.52i)22-s + (1.85 + 1.07i)23-s + ⋯
L(s)  = 1  + (−0.951 + 0.306i)2-s + (0.811 − 0.584i)4-s + (−0.644 + 1.11i)5-s + (−0.997 + 0.0733i)7-s + (−0.593 + 0.805i)8-s + (0.270 − 1.26i)10-s + (−0.878 − 1.52i)11-s + 0.290·13-s + (0.926 − 0.375i)14-s + (0.317 − 0.948i)16-s + (1.43 − 0.827i)17-s + (−0.135 − 0.0781i)19-s + (0.129 + 1.28i)20-s + (1.30 + 1.17i)22-s + (0.387 + 0.223i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.351 + 0.936i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.351 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.361584 - 0.250539i\)
\(L(\frac12)\) \(\approx\) \(0.361584 - 0.250539i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 - 0.434i)T \)
3 \( 1 \)
7 \( 1 + (2.63 - 0.194i)T \)
good5 \( 1 + (1.44 - 2.49i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.91 + 5.04i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.04T + 13T^{2} \)
17 \( 1 + (-5.91 + 3.41i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.589 + 0.340i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.85 - 1.07i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.61iT - 29T^{2} \)
31 \( 1 + (1.91 + 3.31i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.06 + 1.19i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.19iT - 41T^{2} \)
43 \( 1 - 1.34T + 43T^{2} \)
47 \( 1 + (-5.52 + 9.57i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.99 - 4.03i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.81 - 3.93i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.63 + 2.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.65 + 11.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.08iT - 71T^{2} \)
73 \( 1 + (-4.88 + 2.82i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (10.9 + 6.32i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.482iT - 83T^{2} \)
89 \( 1 + (10.7 + 6.19i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.63iT - 97T^{2} \)
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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

−10.66457076610606672523666597188, −9.916455580508406269021003236439, −8.964661783006329427325291847748, −7.88609529300142932721665063562, −7.33516566294916703695971317172, −6.26332715959994796421561799616, −5.58379736418956880401876885001, −3.40025247844433073824739078078, −2.80203112405146211891371496165, −0.37702558780191568144246520831, 1.35932095694777855999447356081, 3.02551812918051071422004578747, 4.19580682376602523784487104511, 5.49887762597015759169823603555, 6.87115579292431913179583676012, 7.67738556102689466362073515223, 8.473471909310679776220102292011, 9.356310712226262160223018646538, 10.10854115763844843317282262013, 10.81565116569965065260337157187

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