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Properties

Label 2-504-56.3-c1-0-15
Degree $2$
Conductor $504$
Sign $0.448 + 0.893i$
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  + (−0.809 − 1.15i)2-s + (−0.690 + 1.87i)4-s + (−1.03 − 1.78i)5-s + (2.39 + 1.13i)7-s + (2.73 − 0.718i)8-s + (−1.23 + 2.64i)10-s + (−0.982 + 1.70i)11-s + 4.20·13-s + (−0.623 − 3.68i)14-s + (−3.04 − 2.59i)16-s + (3.09 + 1.78i)17-s + (−4.36 + 2.52i)19-s + (4.06 − 0.703i)20-s + (2.76 − 0.237i)22-s + (5.31 − 3.06i)23-s + ⋯
L(s)  = 1  + (−0.572 − 0.820i)2-s + (−0.345 + 0.938i)4-s + (−0.461 − 0.799i)5-s + (0.904 + 0.427i)7-s + (0.967 − 0.253i)8-s + (−0.391 + 0.835i)10-s + (−0.296 + 0.512i)11-s + 1.16·13-s + (−0.166 − 0.986i)14-s + (−0.761 − 0.647i)16-s + (0.751 + 0.434i)17-s + (−1.00 + 0.578i)19-s + (0.909 − 0.157i)20-s + (0.590 − 0.0506i)22-s + (1.10 − 0.639i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.448 + 0.893i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.448 + 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.931802 - 0.574728i\)
\(L(\frac12)\) \(\approx\) \(0.931802 - 0.574728i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 1.15i)T \)
3 \( 1 \)
7 \( 1 + (-2.39 - 1.13i)T \)
good5 \( 1 + (1.03 + 1.78i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.982 - 1.70i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.20T + 13T^{2} \)
17 \( 1 + (-3.09 - 1.78i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.36 - 2.52i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.31 + 3.06i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.34iT - 29T^{2} \)
31 \( 1 + (-3.42 + 5.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.56 + 2.05i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.45iT - 41T^{2} \)
43 \( 1 - 4.33T + 43T^{2} \)
47 \( 1 + (-4.88 - 8.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.2 + 6.48i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.17 - 3.56i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.03 + 1.78i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.77 + 4.80i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.44iT - 71T^{2} \)
73 \( 1 + (10.7 + 6.18i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.778 + 0.449i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.58iT - 83T^{2} \)
89 \( 1 + (12.6 - 7.30i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.550iT - 97T^{2} \)
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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

−10.90785667636636180196252892382, −9.923426150758295550980545160105, −8.851472059999264456466404874687, −8.273292990552404124959636948529, −7.67734300197916367452901604830, −6.05360200746216651773774657082, −4.68915194365183972485784197073, −3.99593825473075488500748038831, −2.39096576426096315722606115822, −1.06850119941359080101040658140, 1.21156661686262824768616300586, 3.21672953722618628824794188699, 4.59613951446954661710866935738, 5.61958378322538345273122310721, 6.77030640970026636507182816752, 7.40937623063580115233868346388, 8.343050171715934206169529709608, 8.976289622016024489191218650590, 10.35481932547739985139791722947, 10.93727180705046263271015369722

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