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Properties

Label 2-504-63.4-c1-0-3
Degree $2$
Conductor $504$
Sign $0.946 + 0.322i$
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  + (−1.72 + 0.133i)3-s − 4.22·5-s + (−2.37 + 1.15i)7-s + (2.96 − 0.460i)9-s + 1.92·11-s + (−0.291 + 0.504i)13-s + (7.29 − 0.562i)15-s + (3.61 − 6.25i)17-s + (2.10 + 3.64i)19-s + (3.95 − 2.31i)21-s + 1.27·23-s + 12.8·25-s + (−5.05 + 1.18i)27-s + (−4.20 − 7.27i)29-s + (0.476 + 0.824i)31-s + ⋯
L(s)  = 1  + (−0.997 + 0.0769i)3-s − 1.88·5-s + (−0.898 + 0.438i)7-s + (0.988 − 0.153i)9-s + 0.581·11-s + (−0.0808 + 0.140i)13-s + (1.88 − 0.145i)15-s + (0.875 − 1.51i)17-s + (0.482 + 0.835i)19-s + (0.862 − 0.506i)21-s + 0.266·23-s + 2.56·25-s + (−0.973 + 0.228i)27-s + (−0.780 − 1.35i)29-s + (0.0855 + 0.148i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.946 + 0.322i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.946 + 0.322i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.563738 - 0.0933880i\)
\(L(\frac12)\) \(\approx\) \(0.563738 - 0.0933880i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 - 0.133i)T \)
7 \( 1 + (2.37 - 1.15i)T \)
good5 \( 1 + 4.22T + 5T^{2} \)
11 \( 1 - 1.92T + 11T^{2} \)
13 \( 1 + (0.291 - 0.504i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.61 + 6.25i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.10 - 3.64i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.27T + 23T^{2} \)
29 \( 1 + (4.20 + 7.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.476 - 0.824i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.03 - 5.25i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.31 + 2.27i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.442 - 0.766i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.88 - 4.99i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.962 - 1.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.27 - 3.94i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.29 + 9.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.43 - 4.21i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.5T + 71T^{2} \)
73 \( 1 + (-0.446 + 0.772i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.93 + 10.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.24 + 9.08i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.87 - 6.71i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.98 + 3.44i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.24752251620071133239241736320, −9.993835846361120890222544113164, −9.264181684218671813758245851408, −7.913213547258601945339693870054, −7.24576134409447971781268276789, −6.32347069440460156903807016932, −5.15531750731836962183228786590, −4.09596258707585516849540996857, −3.21775095225591091225620571640, −0.61925304214287558081790021112, 0.834125618975644538502094742088, 3.49206197427613962359615573876, 4.05237893157568640823701596594, 5.28153311831539665158339296807, 6.57106674086578335875774279151, 7.21878569267140984092937428039, 8.033354262772195385256844468667, 9.233838407655682674814245353511, 10.38466328387851304810678129774, 11.08193052222446449395890791644

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