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Properties

Label 2-504-168.125-c1-0-16
Degree $2$
Conductor $504$
Sign $0.758 + 0.651i$
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  + (−1.39 + 0.245i)2-s + (1.87 − 0.684i)4-s − 2.37i·5-s + (−1.53 + 2.15i)7-s + (−2.44 + 1.41i)8-s + (0.582 + 3.30i)10-s − 0.335·11-s + 6.30·13-s + (1.60 − 3.38i)14-s + (3.06 − 2.57i)16-s + 4.10·17-s − 2.19·19-s + (−1.62 − 4.45i)20-s + (0.467 − 0.0825i)22-s − 6.23i·23-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s − 1.06i·5-s + (−0.579 + 0.815i)7-s + (−0.866 + 0.500i)8-s + (0.184 + 1.04i)10-s − 0.101·11-s + 1.74·13-s + (0.428 − 0.903i)14-s + (0.766 − 0.642i)16-s + 0.996·17-s − 0.502·19-s + (−0.362 − 0.997i)20-s + (0.0997 − 0.0175i)22-s − 1.30i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.758 + 0.651i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.758 + 0.651i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.841728 - 0.311781i\)
\(L(\frac12)\) \(\approx\) \(0.841728 - 0.311781i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.245i)T \)
3 \( 1 \)
7 \( 1 + (1.53 - 2.15i)T \)
good5 \( 1 + 2.37iT - 5T^{2} \)
11 \( 1 + 0.335T + 11T^{2} \)
13 \( 1 - 6.30T + 13T^{2} \)
17 \( 1 - 4.10T + 17T^{2} \)
19 \( 1 + 2.19T + 19T^{2} \)
23 \( 1 + 6.23iT - 23T^{2} \)
29 \( 1 + 4.38T + 29T^{2} \)
31 \( 1 + 8.10iT - 31T^{2} \)
37 \( 1 + 9.97iT - 37T^{2} \)
41 \( 1 - 7.35T + 41T^{2} \)
43 \( 1 + 0.892iT - 43T^{2} \)
47 \( 1 - 9.34T + 47T^{2} \)
53 \( 1 - 0.748T + 53T^{2} \)
59 \( 1 - 8.91iT - 59T^{2} \)
61 \( 1 - 7.33T + 61T^{2} \)
67 \( 1 - 5.56iT - 67T^{2} \)
71 \( 1 - 2.08iT - 71T^{2} \)
73 \( 1 - 5.81iT - 73T^{2} \)
79 \( 1 + 9.27T + 79T^{2} \)
83 \( 1 + 1.64iT - 83T^{2} \)
89 \( 1 + 11.3T + 89T^{2} \)
97 \( 1 + 2.81iT - 97T^{2} \)
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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

−10.70865951207903742872091761645, −9.705724293704401414702317789262, −8.790605249030757470686938952891, −8.564597139197568138113134197299, −7.39353823746265804341769865970, −6.02464100029709629682374299058, −5.67465436986893848568902326511, −3.94925665758982682912004200522, −2.39687265760816326844980081349, −0.862230824515607439777156597609, 1.31680446716978259799227716677, 3.10013145736337110641534771711, 3.70448443991613948425840795126, 5.86510839064777529873863404308, 6.67130729906900927271107523824, 7.41154070917040178042264079294, 8.324685848066119283344683989377, 9.386366506204796433236370881366, 10.25903509003022472063368180313, 10.82163783124953484520403518467

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