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Properties

Label 2-504-56.13-c2-0-11
Degree $2$
Conductor $504$
Sign $-0.916 - 0.399i$
Analytic cond. $13.7330$
Root an. cond. $3.70580$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.44 + 1.38i)2-s + (0.187 − 3.99i)4-s + 5.24·5-s + (−6.31 + 3.01i)7-s + (5.24 + 6.03i)8-s + (−7.58 + 7.24i)10-s − 3.67i·11-s − 12.5·13-s + (4.98 − 13.0i)14-s + (−15.9 − 1.49i)16-s + 18.2i·17-s + 4.94·19-s + (0.982 − 20.9i)20-s + (5.08 + 5.32i)22-s − 1.93·23-s + ⋯
L(s)  = 1  + (−0.723 + 0.690i)2-s + (0.0468 − 0.998i)4-s + 1.04·5-s + (−0.902 + 0.430i)7-s + (0.655 + 0.754i)8-s + (−0.758 + 0.724i)10-s − 0.334i·11-s − 0.962·13-s + (0.356 − 0.934i)14-s + (−0.995 − 0.0935i)16-s + 1.07i·17-s + 0.260·19-s + (0.0491 − 1.04i)20-s + (0.230 + 0.242i)22-s − 0.0841·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 - 0.399i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.916 - 0.399i$
Analytic conductor: \(13.7330\)
Root analytic conductor: \(3.70580\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1),\ -0.916 - 0.399i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7013916638\)
\(L(\frac12)\) \(\approx\) \(0.7013916638\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.44 - 1.38i)T \)
3 \( 1 \)
7 \( 1 + (6.31 - 3.01i)T \)
good5 \( 1 - 5.24T + 25T^{2} \)
11 \( 1 + 3.67iT - 121T^{2} \)
13 \( 1 + 12.5T + 169T^{2} \)
17 \( 1 - 18.2iT - 289T^{2} \)
19 \( 1 - 4.94T + 361T^{2} \)
23 \( 1 + 1.93T + 529T^{2} \)
29 \( 1 - 41.0iT - 841T^{2} \)
31 \( 1 + 22.7iT - 961T^{2} \)
37 \( 1 - 46.7iT - 1.36e3T^{2} \)
41 \( 1 - 66.3iT - 1.68e3T^{2} \)
43 \( 1 - 41.9iT - 1.84e3T^{2} \)
47 \( 1 + 53.3iT - 2.20e3T^{2} \)
53 \( 1 - 61.3iT - 2.80e3T^{2} \)
59 \( 1 + 70.9T + 3.48e3T^{2} \)
61 \( 1 + 82.0T + 3.72e3T^{2} \)
67 \( 1 - 1.63iT - 4.48e3T^{2} \)
71 \( 1 + 10.0T + 5.04e3T^{2} \)
73 \( 1 + 64.4iT - 5.32e3T^{2} \)
79 \( 1 + 149.T + 6.24e3T^{2} \)
83 \( 1 + 12.9T + 6.88e3T^{2} \)
89 \( 1 + 130. iT - 7.92e3T^{2} \)
97 \( 1 - 103. iT - 9.40e3T^{2} \)
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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

−10.66832198193618573285379584261, −9.926892508086211449344735276068, −9.403827244754181873309476077822, −8.535605333715912743304753488304, −7.46018323519720050381909095031, −6.34376147409233302367051459485, −5.91932316128503660656854494337, −4.81542476577998182041184790314, −2.91493058438235314973149254813, −1.58885800442044509504418990054, 0.34014581711342266368909695966, 2.02812251570958769917504292898, 2.98525349104474560162958694265, 4.33354346249538887750053523058, 5.69072131282967998557241869373, 6.93734990088724504572378318717, 7.53344747007927108955340054497, 8.955892649719856317132212632725, 9.691736083100640932462008910819, 10.01245130656498831237333661273

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