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Properties

Label 2-504-168.125-c1-0-0
Degree $2$
Conductor $504$
Sign $-0.864 - 0.501i$
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 + (2.66 − 2.62i)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 + (0.711 − 0.702i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 - 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.864 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.864 - 0.501i$
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: \((2,\ 504,\ (\ :1/2),\ -0.864 - 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0141108 + 0.0524404i\)
\(L(\frac12)\) \(\approx\) \(0.0141108 + 0.0524404i\)
\(L(\frac{3}{2})\) 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

−11.45795901052894934239718895206, −9.953678061111467160455551761929, −9.532751077348975201358373210723, −8.807427266803926903933214560740, −7.87722087807457500413711143962, −6.96412098427433669383579799478, −5.77845917538411262539377631470, −4.76024387544966242158859629573, −3.06979011612730893651334094104, −1.87094385558667432888727710234, 0.04035743719286958335126563270, 2.25859250299897444183908290765, 3.28021749073570367992804418207, 4.93612248022616867880245668169, 6.44460441963500520921028739820, 7.01482121639026494785156060670, 7.60218105001330054716416340223, 8.882858003665265567755826110736, 9.809521784637933200515004893884, 10.43974306473651667842655466837

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