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Properties

Label 2-504-56.3-c1-0-24
Degree $2$
Conductor $504$
Sign $0.599 - 0.800i$
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  + (1.30 + 0.554i)2-s + (1.38 + 1.44i)4-s + (0.345 + 0.597i)5-s + (2.63 + 0.222i)7-s + (1.00 + 2.64i)8-s + (0.117 + 0.969i)10-s + (1.63 − 2.82i)11-s − 5.27·13-s + (3.30 + 1.75i)14-s + (−0.167 + 3.99i)16-s + (2.20 + 1.27i)17-s + (−0.484 + 0.279i)19-s + (−0.385 + 1.32i)20-s + (3.68 − 2.76i)22-s + (−2.50 + 1.44i)23-s + ⋯
L(s)  = 1  + (0.919 + 0.392i)2-s + (0.692 + 0.721i)4-s + (0.154 + 0.267i)5-s + (0.996 + 0.0840i)7-s + (0.353 + 0.935i)8-s + (0.0370 + 0.306i)10-s + (0.491 − 0.851i)11-s − 1.46·13-s + (0.883 + 0.468i)14-s + (−0.0417 + 0.999i)16-s + (0.534 + 0.308i)17-s + (−0.111 + 0.0642i)19-s + (−0.0861 + 0.296i)20-s + (0.786 − 0.590i)22-s + (−0.522 + 0.301i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.599 - 0.800i$
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: \((2,\ 504,\ (\ :1/2),\ 0.599 - 0.800i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.39846 + 1.19967i\)
\(L(\frac12)\) \(\approx\) \(2.39846 + 1.19967i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30 - 0.554i)T \)
3 \( 1 \)
7 \( 1 + (-2.63 - 0.222i)T \)
good5 \( 1 + (-0.345 - 0.597i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.63 + 2.82i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.27T + 13T^{2} \)
17 \( 1 + (-2.20 - 1.27i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.484 - 0.279i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.50 - 1.44i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.444iT - 29T^{2} \)
31 \( 1 + (4.45 - 7.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.00 + 3.46i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.76iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-2.20 - 3.81i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.17 + 4.71i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (8.59 + 4.96i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.23 + 9.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.45 - 2.51i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.29iT - 71T^{2} \)
73 \( 1 + (5.28 + 3.05i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.01 + 2.89i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.83iT - 83T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.42iT - 97T^{2} \)
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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

−11.17299063964792143127830779171, −10.47428306799914325275094847857, −9.115967990348891327233117386158, −8.068081137721308218909779164783, −7.37928097433444601372607338226, −6.29643372604631269766517368915, −5.37075351838768075426719740669, −4.50016354749647425896901898005, −3.27779596138057874068582937038, −1.99983118682972407606865484212, 1.52565051987157310996247378648, 2.67728737996027828370331919080, 4.29808354456778946397516444632, 4.85261581822579691484141070160, 5.85810912058314718269784127633, 7.14092658243805035443580486555, 7.79439445319447056593983863399, 9.356826917121554140782743789268, 9.953197599415778397132361059862, 11.02659431655391913594122037555

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