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Properties

Label 2-504-56.53-c1-0-0
Degree $2$
Conductor $504$
Sign $-0.961 - 0.275i$
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 − 0.190i)2-s + (1.92 + 0.535i)4-s + (−1.93 + 1.11i)5-s + (−0.5 − 2.59i)7-s + (−2.59 − 1.11i)8-s + (2.92 − 1.19i)10-s + (1.93 + 1.11i)11-s + (0.204 + 3.73i)14-s + (3.42 + 2.06i)16-s + (−1.73 + 3i)17-s + (−6.70 + 3.87i)19-s + (−4.33 + 1.11i)20-s + (−2.5 − 1.93i)22-s + (−3.46 − 6i)23-s + (0.427 − 5.27i)28-s − 2.23i·29-s + ⋯
L(s)  = 1  + (−0.990 − 0.135i)2-s + (0.963 + 0.267i)4-s + (−0.866 + 0.499i)5-s + (−0.188 − 0.981i)7-s + (−0.918 − 0.395i)8-s + (0.925 − 0.378i)10-s + (0.583 + 0.337i)11-s + (0.0546 + 0.998i)14-s + (0.856 + 0.515i)16-s + (−0.420 + 0.727i)17-s + (−1.53 + 0.888i)19-s + (−0.968 + 0.249i)20-s + (−0.533 − 0.412i)22-s + (−0.722 − 1.25i)23-s + (0.0807 − 0.996i)28-s − 0.415i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.961 - 0.275i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.961 - 0.275i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0108077 + 0.0767968i\)
\(L(\frac12)\) \(\approx\) \(0.0108077 + 0.0767968i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.190i)T \)
3 \( 1 \)
7 \( 1 + (0.5 + 2.59i)T \)
good5 \( 1 + (1.93 - 1.11i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.93 - 1.11i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (1.73 - 3i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.70 - 3.87i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.46 + 6i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.23iT - 29T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.70 - 3.87i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (-1.73 - 3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.68 + 5.59i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.93 - 1.11i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.70 - 3.87i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.70 + 3.87i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.1iT - 83T^{2} \)
89 \( 1 + (-6.92 - 12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−10.95744439647818109885761564928, −10.52926616570390709193257512479, −9.707672858306954366417835820357, −8.422148346935828612386557947908, −7.950589333590036213271410092792, −6.79416467854633837547311889909, −6.39430119970156005652205823175, −4.26331213772613422192347401540, −3.50118743199213450526494886889, −1.85770559900006643034437105861, 0.06014735854066869138333251280, 1.98848909123764915009343502910, 3.41650811721813473138876695361, 4.89871876625288706827767329800, 6.10311615598144891661191286538, 6.97829994312082165138254670831, 8.062980722973341318719293925879, 8.815227683163330592698236095269, 9.264110714953672409564107922533, 10.47727969085068565199850894991

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