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Properties

Label 2-504-24.11-c1-0-10
Degree $2$
Conductor $504$
Sign $0.657 + 0.753i$
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  + (−1.40 − 0.111i)2-s + (1.97 + 0.315i)4-s − 0.472·5-s + i·7-s + (−2.74 − 0.665i)8-s + (0.665 + 0.0528i)10-s − 3.86i·11-s − 3.00i·13-s + (0.111 − 1.40i)14-s + (3.80 + 1.24i)16-s + 1.23i·17-s + 6.30·19-s + (−0.932 − 0.148i)20-s + (−0.431 + 5.44i)22-s + 1.60·23-s + ⋯
L(s)  = 1  + (−0.996 − 0.0790i)2-s + (0.987 + 0.157i)4-s − 0.211·5-s + 0.377i·7-s + (−0.971 − 0.235i)8-s + (0.210 + 0.0167i)10-s − 1.16i·11-s − 0.832i·13-s + (0.0298 − 0.376i)14-s + (0.950 + 0.311i)16-s + 0.298i·17-s + 1.44·19-s + (−0.208 − 0.0333i)20-s + (−0.0920 + 1.16i)22-s + 0.333·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.657 + 0.753i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.657 + 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.779634 - 0.354228i\)
\(L(\frac12)\) \(\approx\) \(0.779634 - 0.354228i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.111i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 + 0.472T + 5T^{2} \)
11 \( 1 + 3.86iT - 11T^{2} \)
13 \( 1 + 3.00iT - 13T^{2} \)
17 \( 1 - 1.23iT - 17T^{2} \)
19 \( 1 - 6.30T + 19T^{2} \)
23 \( 1 - 1.60T + 23T^{2} \)
29 \( 1 - 6.36T + 29T^{2} \)
31 \( 1 + 9.09iT - 31T^{2} \)
37 \( 1 - 0.844iT - 37T^{2} \)
41 \( 1 + 2.07iT - 41T^{2} \)
43 \( 1 - 5.97T + 43T^{2} \)
47 \( 1 - 7.22T + 47T^{2} \)
53 \( 1 - 6.64T + 53T^{2} \)
59 \( 1 + 7.22iT - 59T^{2} \)
61 \( 1 + 7.67iT - 61T^{2} \)
67 \( 1 - 0.304T + 67T^{2} \)
71 \( 1 - 4.28T + 71T^{2} \)
73 \( 1 + 11.9T + 73T^{2} \)
79 \( 1 + 8.86iT - 79T^{2} \)
83 \( 1 + 9.28iT - 83T^{2} \)
89 \( 1 - 16.1iT - 89T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−10.74314278061528621073069617899, −9.851438790039926006556166565793, −9.009849932240431942465549450132, −8.122558210142584164779507954018, −7.52279473275543973565017171663, −6.20715116508816072542249386216, −5.48682473313089605622900589120, −3.62241198660568209985464372305, −2.59499287676198874740504607207, −0.802021335988745910599356234418, 1.34113099036640907622714555457, 2.79440735751896678228708774837, 4.29353869592763574425990168220, 5.57643280089180198689409996196, 7.00511527835453691611155916688, 7.25116136330889386097830994516, 8.411223447715267352035693733438, 9.373383417745409331367286291649, 9.985963952390293937117896652639, 10.86126209476705275359432317640

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