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Properties

Label 2-504-8.5-c1-0-3
Degree $2$
Conductor $504$
Sign $-0.947 - 0.321i$
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  + (0.569 + 1.29i)2-s + (−1.35 + 1.47i)4-s − 0.772i·5-s − 7-s + (−2.67 − 0.908i)8-s + (1 − 0.440i)10-s + 4.40i·11-s + 5.89i·13-s + (−0.569 − 1.29i)14-s + (−0.350 − 3.98i)16-s − 4.21·17-s + 5.01i·19-s + (1.13 + 1.04i)20-s + (−5.70 + 2.50i)22-s − 8.77·23-s + ⋯
L(s)  = 1  + (0.402 + 0.915i)2-s + (−0.675 + 0.737i)4-s − 0.345i·5-s − 0.377·7-s + (−0.947 − 0.321i)8-s + (0.316 − 0.139i)10-s + 1.32i·11-s + 1.63i·13-s + (−0.152 − 0.345i)14-s + (−0.0876 − 0.996i)16-s − 1.02·17-s + 1.15i·19-s + (0.254 + 0.233i)20-s + (−1.21 + 0.535i)22-s − 1.82·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.321i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.947 - 0.321i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.947 - 0.321i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.189130 + 1.14699i\)
\(L(\frac12)\) \(\approx\) \(0.189130 + 1.14699i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.569 - 1.29i)T \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 + 0.772iT - 5T^{2} \)
11 \( 1 - 4.40iT - 11T^{2} \)
13 \( 1 - 5.89iT - 13T^{2} \)
17 \( 1 + 4.21T + 17T^{2} \)
19 \( 1 - 5.01iT - 19T^{2} \)
23 \( 1 + 8.77T + 23T^{2} \)
29 \( 1 + 3.63iT - 29T^{2} \)
31 \( 1 - 7.40T + 31T^{2} \)
37 \( 1 + 5.01iT - 37T^{2} \)
41 \( 1 - 8.77T + 41T^{2} \)
43 \( 1 - 0.880iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 6.72iT - 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 - 5.89iT - 61T^{2} \)
67 \( 1 - 0.880iT - 67T^{2} \)
71 \( 1 - 4.21T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 1.54iT - 83T^{2} \)
89 \( 1 + 8.77T + 89T^{2} \)
97 \( 1 - 16.8T + 97T^{2} \)
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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

−11.70095511905992171704956464313, −10.11358065763086707652124936076, −9.403260796991431556016738390853, −8.557579740085972405309251209741, −7.53989093201416320723350879201, −6.67856539026713312839643721224, −5.92868703716718867277427860763, −4.47220528883156926437514537534, −4.12964996038375060231337356952, −2.20606999834355670500158393509, 0.59991832831527631989140407058, 2.62560780960091087840788476497, 3.33390566448038367971146836811, 4.62818540597513655800815029695, 5.76993003600210567870097874666, 6.51524852842085237742132249772, 8.082854423207401099519156653233, 8.852020349872353390366423313131, 9.919847905374537481671116966827, 10.72223359111325229108657533998

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