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Properties

Label 2-504-63.41-c1-0-13
Degree $2$
Conductor $504$
Sign $0.915 - 0.402i$
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  + (1.73 + 0.0591i)3-s + (−1.11 + 1.92i)5-s + (1.78 − 1.95i)7-s + (2.99 + 0.204i)9-s + (−1.51 + 0.876i)11-s + (3.37 + 1.94i)13-s + (−2.04 + 3.27i)15-s + 1.78·17-s − 2.73i·19-s + (3.20 − 3.27i)21-s + (1.64 + 0.947i)23-s + (0.0196 + 0.0339i)25-s + (5.16 + 0.531i)27-s + (−7.48 + 4.32i)29-s + (7.18 + 4.14i)31-s + ⋯
L(s)  = 1  + (0.999 + 0.0341i)3-s + (−0.498 + 0.862i)5-s + (0.673 − 0.738i)7-s + (0.997 + 0.0682i)9-s + (−0.457 + 0.264i)11-s + (0.935 + 0.540i)13-s + (−0.527 + 0.845i)15-s + 0.433·17-s − 0.628i·19-s + (0.698 − 0.715i)21-s + (0.342 + 0.197i)23-s + (0.00392 + 0.00678i)25-s + (0.994 + 0.102i)27-s + (−1.39 + 0.802i)29-s + (1.29 + 0.745i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.915 - 0.402i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.915 - 0.402i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97057 + 0.413683i\)
\(L(\frac12)\) \(\approx\) \(1.97057 + 0.413683i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.73 - 0.0591i)T \)
7 \( 1 + (-1.78 + 1.95i)T \)
good5 \( 1 + (1.11 - 1.92i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.51 - 0.876i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.37 - 1.94i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.78T + 17T^{2} \)
19 \( 1 + 2.73iT - 19T^{2} \)
23 \( 1 + (-1.64 - 0.947i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.48 - 4.32i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.18 - 4.14i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 9.91T + 37T^{2} \)
41 \( 1 + (-3.42 + 5.93i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.22 + 7.32i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.47 + 2.54i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.45iT - 53T^{2} \)
59 \( 1 + (-0.449 + 0.778i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.3 - 5.98i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.38 + 9.32i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.2iT - 71T^{2} \)
73 \( 1 + 4.71iT - 73T^{2} \)
79 \( 1 + (4.70 + 8.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.326 - 0.565i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.70T + 89T^{2} \)
97 \( 1 + (-0.0294 + 0.0169i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.79803677806319804328469119189, −10.27716726061869289964074235246, −9.034464987486649886818699641195, −8.278167399668973420380002540283, −7.26418512673384700911075768641, −6.91403589655129828452173902952, −5.11988633871720222459696563577, −3.93086052451269382184902010172, −3.17699872066074719464271249950, −1.68502848923533743083808336958, 1.38270190578060188187706447361, 2.84613963759032445149675905633, 4.02978141800282102307002532622, 5.05648965547159934605982038035, 6.15088317870482725467593231987, 7.81771687123035863635864604316, 8.124317583783702656942232768473, 8.850590436005574329660660025797, 9.765909355264290629089544320816, 10.86987307289984057369284661122

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