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Properties

Label 2-504-63.20-c1-0-16
Degree $2$
Conductor $504$
Sign $0.0426 + 0.999i$
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  + (0.0936 − 1.72i)3-s + (1.26 + 2.19i)5-s + (0.521 − 2.59i)7-s + (−2.98 − 0.323i)9-s + (−5.68 − 3.28i)11-s + (5.13 − 2.96i)13-s + (3.91 − 1.98i)15-s + 3.52·17-s − 0.261i·19-s + (−4.43 − 1.14i)21-s + (2.89 − 1.67i)23-s + (−0.718 + 1.24i)25-s + (−0.839 + 5.12i)27-s + (−1.00 − 0.582i)29-s + (1.69 − 0.977i)31-s + ⋯
L(s)  = 1  + (0.0540 − 0.998i)3-s + (0.567 + 0.982i)5-s + (0.196 − 0.980i)7-s + (−0.994 − 0.107i)9-s + (−1.71 − 0.989i)11-s + (1.42 − 0.822i)13-s + (1.01 − 0.513i)15-s + 0.855·17-s − 0.0599i·19-s + (−0.968 − 0.249i)21-s + (0.604 − 0.349i)23-s + (−0.143 + 0.248i)25-s + (−0.161 + 0.986i)27-s + (−0.187 − 0.108i)29-s + (0.304 − 0.175i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0426 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0426 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.0426 + 0.999i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.0426 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05944 - 1.01522i\)
\(L(\frac12)\) \(\approx\) \(1.05944 - 1.01522i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.0936 + 1.72i)T \)
7 \( 1 + (-0.521 + 2.59i)T \)
good5 \( 1 + (-1.26 - 2.19i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.68 + 3.28i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.13 + 2.96i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.52T + 17T^{2} \)
19 \( 1 + 0.261iT - 19T^{2} \)
23 \( 1 + (-2.89 + 1.67i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.00 + 0.582i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.69 + 0.977i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.16T + 37T^{2} \)
41 \( 1 + (2.85 + 4.94i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.67 + 4.63i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.79 - 6.57i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.54iT - 53T^{2} \)
59 \( 1 + (-5.47 - 9.48i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.53 + 3.19i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.54 - 7.86i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.5iT - 71T^{2} \)
73 \( 1 - 2.72iT - 73T^{2} \)
79 \( 1 + (0.652 - 1.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.53 - 7.85i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + (-9.95 - 5.74i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.71689475944636862304811773382, −10.22419249163124926122425124237, −8.573633564937325591949546407215, −7.905523551556094112256480540005, −7.11841674341708198369994034786, −6.09298893432406503528299058332, −5.43459094810006750823512721116, −3.43701919333014361953793691102, −2.64265018621634394298201483319, −0.914978538705203489013723100707, 1.88861465355617269023336847247, 3.27616507905520644292856155934, 4.81799479800596475661214228922, 5.21137881826358114890596286224, 6.16446539470619380553754110293, 7.88527438188258573699858807299, 8.662142783833508319077281946079, 9.360585534602877373504123808919, 10.09008364847000303286117367504, 11.03580736626246923015637089826

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