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Properties

Label 2-504-24.11-c1-0-11
Degree $2$
Conductor $504$
Sign $0.462 - 0.886i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.462 - 0.886i$
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: \((2,\ 504,\ (\ :1/2),\ 0.462 - 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.26907 + 1.37501i\)
\(L(\frac12)\) \(\approx\) \(2.26907 + 1.37501i\)
\(L(\frac{3}{2})\) 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

−11.20738261298636331464373572247, −10.27592120590462635120267034283, −9.187586410757883567781826175809, −8.310028839368980714138158552144, −7.14989956818722497696926575768, −6.37307585455365976091686017743, −5.41116785129729277400975466551, −4.64487243864502104054177391790, −3.18311935105330564119974670647, −2.10013219618323962515962939956, 1.46478940676958177600439419244, 2.72494322449955764532006006947, 3.99595200700096760560916896089, 4.97931156365418416459719287068, 6.11794832040098528460609632872, 6.63843537106439685916755077865, 8.039032966430050892536899310663, 9.256850327422641732074259065572, 10.16469882836852144021945642145, 10.75108351643732047852548821110

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