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Properties

Label 2-504-56.27-c1-0-15
Degree $2$
Conductor $504$
Sign $-0.0716 - 0.997i$
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 + i)2-s + 2i·4-s + 3.46·5-s + (−1.73 + 2i)7-s + (−2 + 2i)8-s + (3.46 + 3.46i)10-s − 2·11-s + 3.46·13-s + (−3.73 + 0.267i)14-s − 4·16-s + 3.46i·17-s − 6.92i·19-s + 6.92i·20-s + (−2 − 2i)22-s − 2i·23-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + i·4-s + 1.54·5-s + (−0.654 + 0.755i)7-s + (−0.707 + 0.707i)8-s + (1.09 + 1.09i)10-s − 0.603·11-s + 0.960·13-s + (−0.997 + 0.0716i)14-s − 16-s + 0.840i·17-s − 1.58i·19-s + 1.54i·20-s + (−0.426 − 0.426i)22-s − 0.417i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0716 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0716 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.0716 - 0.997i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.0716 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61634 + 1.73655i\)
\(L(\frac12)\) \(\approx\) \(1.61634 + 1.73655i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 \)
7 \( 1 + (1.73 - 2i)T \)
good5 \( 1 - 3.46T + 5T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 - 3.46iT - 17T^{2} \)
19 \( 1 + 6.92iT - 19T^{2} \)
23 \( 1 + 2iT - 23T^{2} \)
29 \( 1 - 8iT - 29T^{2} \)
31 \( 1 - 3.46T + 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 - 4iT - 53T^{2} \)
59 \( 1 - 6.92iT - 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 - 10T + 67T^{2} \)
71 \( 1 - 2iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 12iT - 79T^{2} \)
83 \( 1 + 6.92iT - 83T^{2} \)
89 \( 1 + 10.3iT - 89T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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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

−11.12789746312879149087340185997, −10.26071603594475396620348973877, −8.976649843788313505550426384224, −8.763285279184989153050754896489, −7.18973410787547233367326601808, −6.23466015417750218056857508454, −5.76889489847241323040371488662, −4.81565402844349154016123286053, −3.22461266833243799297888382303, −2.22841257062652482863833157614, 1.29168738271627690120488975094, 2.59893755589607865092552444958, 3.71385986925118541748507299125, 4.99909179566428918993886433152, 6.03518881728731297856626236946, 6.46536442656294721357299602826, 8.013640430106141172242941288110, 9.571254280974355735709968044641, 9.820300247458261435513703606238, 10.59507753691665026707017781771

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