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Properties

Label 2-504-24.11-c1-0-19
Degree $2$
Conductor $504$
Sign $-0.681 + 0.731i$
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  + (−1.25 + 0.651i)2-s + (1.15 − 1.63i)4-s − 1.82·5-s − i·7-s + (−0.378 + 2.80i)8-s + (2.29 − 1.18i)10-s + 0.868i·11-s + 0.873i·13-s + (0.651 + 1.25i)14-s + (−1.35 − 3.76i)16-s − 4.00i·17-s − 3.39·19-s + (−2.09 + 2.98i)20-s + (−0.565 − 1.09i)22-s − 3.82·23-s + ⋯
L(s)  = 1  + (−0.887 + 0.460i)2-s + (0.575 − 0.817i)4-s − 0.816·5-s − 0.377i·7-s + (−0.133 + 0.990i)8-s + (0.724 − 0.375i)10-s + 0.261i·11-s + 0.242i·13-s + (0.174 + 0.335i)14-s + (−0.337 − 0.941i)16-s − 0.972i·17-s − 0.778·19-s + (−0.469 + 0.667i)20-s + (−0.120 − 0.232i)22-s − 0.798·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.681 + 0.731i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.681 + 0.731i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0821223 - 0.188682i\)
\(L(\frac12)\) \(\approx\) \(0.0821223 - 0.188682i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.25 - 0.651i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 + 1.82T + 5T^{2} \)
11 \( 1 - 0.868iT - 11T^{2} \)
13 \( 1 - 0.873iT - 13T^{2} \)
17 \( 1 + 4.00iT - 17T^{2} \)
19 \( 1 + 3.39T + 19T^{2} \)
23 \( 1 + 3.82T + 23T^{2} \)
29 \( 1 + 7.14T + 29T^{2} \)
31 \( 1 + 4.04iT - 31T^{2} \)
37 \( 1 - 2.16iT - 37T^{2} \)
41 \( 1 + 8.89iT - 41T^{2} \)
43 \( 1 + 9.10T + 43T^{2} \)
47 \( 1 + 7.42T + 47T^{2} \)
53 \( 1 - 9.49T + 53T^{2} \)
59 \( 1 + 7.42iT - 59T^{2} \)
61 \( 1 - 2.29iT - 61T^{2} \)
67 \( 1 + 12.6T + 67T^{2} \)
71 \( 1 + 3.29T + 71T^{2} \)
73 \( 1 - 15.2T + 73T^{2} \)
79 \( 1 + 3.11iT - 79T^{2} \)
83 \( 1 + 0.0325iT - 83T^{2} \)
89 \( 1 + 13.6iT - 89T^{2} \)
97 \( 1 + 3.79T + 97T^{2} \)
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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

−10.44046527667460298902836356952, −9.650120513590567907237380835166, −8.731169331476640093656699322480, −7.80105816588502739326616920062, −7.21631111378544915148888524065, −6.21922512100671290922153397093, −4.98425226023319928092124345678, −3.77173631131762934444550473085, −2.05528666381652873441058480553, −0.15599738526607778296338962857, 1.82819335555489827517152223412, 3.28101052807844608981697629022, 4.20517333209737624390303771846, 5.86307609122199796659040065168, 6.91986105323144092189129379693, 8.049494605159421751049153285685, 8.401897320611762184837223551489, 9.492494429392368101406636337805, 10.40501441684596939546472857946, 11.18816586814542491415946324243

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