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Properties

Label 2-504-56.3-c1-0-5
Degree $2$
Conductor $504$
Sign $-0.0235 - 0.999i$
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  + (−1.13 − 0.849i)2-s + (0.557 + 1.92i)4-s + (1.59 + 2.76i)5-s + (−0.694 + 2.55i)7-s + (1.00 − 2.64i)8-s + (0.542 − 4.48i)10-s + (−0.800 + 1.38i)11-s + 1.38·13-s + (2.95 − 2.29i)14-s + (−3.37 + 2.14i)16-s + (−3.48 − 2.01i)17-s + (−4.56 + 2.63i)19-s + (−4.42 + 4.61i)20-s + (2.08 − 0.888i)22-s + (−3.83 + 2.21i)23-s + ⋯
L(s)  = 1  + (−0.799 − 0.600i)2-s + (0.278 + 0.960i)4-s + (0.714 + 1.23i)5-s + (−0.262 + 0.964i)7-s + (0.353 − 0.935i)8-s + (0.171 − 1.41i)10-s + (−0.241 + 0.418i)11-s + 0.385·13-s + (0.789 − 0.614i)14-s + (−0.844 + 0.535i)16-s + (−0.845 − 0.488i)17-s + (−1.04 + 0.605i)19-s + (−0.988 + 1.03i)20-s + (0.444 − 0.189i)22-s + (−0.798 + 0.461i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0235 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0235 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.0235 - 0.999i$
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: \((2,\ 504,\ (\ :1/2),\ -0.0235 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.588231 + 0.602226i\)
\(L(\frac12)\) \(\approx\) \(0.588231 + 0.602226i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13 + 0.849i)T \)
3 \( 1 \)
7 \( 1 + (0.694 - 2.55i)T \)
good5 \( 1 + (-1.59 - 2.76i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.800 - 1.38i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.38T + 13T^{2} \)
17 \( 1 + (3.48 + 2.01i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.56 - 2.63i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.83 - 2.21i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.10iT - 29T^{2} \)
31 \( 1 + (-0.0579 + 0.100i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.63 + 2.67i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.21iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-5.05 - 8.76i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.13 - 3.54i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.38 - 2.53i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.21 - 7.29i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.01 + 8.69i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.29iT - 71T^{2} \)
73 \( 1 + (-9.30 - 5.37i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.3 - 5.96i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.9iT - 83T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.87iT - 97T^{2} \)
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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

−11.02224663376967936118737337913, −10.17598647144315995550496416266, −9.579392015328024083227766509705, −8.665835491337125180621450540955, −7.64905744494781785715195881177, −6.58338029689523042415590875861, −5.91361952366068390920058635114, −4.09606281138815564564131487621, −2.70184217697430889809419679483, −2.12199384444095951122335904591, 0.62771475276367029318442108251, 2.01550256147075625645287827576, 4.17430848331995248055302044593, 5.20492832330551522113677853505, 6.21154729799317109659483924952, 6.99838291221786992074319592271, 8.353710331274755311052781841805, 8.699668173570238174941223466816, 9.699542012604970536442808308669, 10.48189005664696768832057620725

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