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Properties

Label 2-504-56.13-c0-0-1
Degree $2$
Conductor $504$
Sign $i$
Analytic cond. $0.251528$
Root an. cond. $0.501526$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − 4-s + 7-s + i·8-s − 2i·11-s − i·14-s + 16-s − 2·22-s − 25-s − 28-s + 2i·29-s − i·32-s + 2i·44-s + 49-s + i·50-s + ⋯
L(s)  = 1  − i·2-s − 4-s + 7-s + i·8-s − 2i·11-s − i·14-s + 16-s − 2·22-s − 25-s − 28-s + 2i·29-s − i·32-s + 2i·44-s + 49-s + i·50-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $i$
Analytic conductor: \(0.251528\)
Root analytic conductor: \(0.501526\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8595064973\)
\(L(\frac12)\) \(\approx\) \(0.8595064973\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + T^{2} \)
11 \( 1 + 2iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - 2iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 2iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 2T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−11.02736288555510689890473445734, −10.35107207482332204488348574358, −9.027915968570364710771121702588, −8.534015845060712995438600939245, −7.61542465395263608247968716370, −5.95012340013087900626517670500, −5.13862652968003523422956471952, −3.92634248655435326341959254362, −2.89421827024905918171286179056, −1.34936159779161659946630610031, 1.98369331142917368553873576877, 4.11112593998298205424011014724, 4.75866191754171657278325285436, 5.79300872100055039029290891748, 6.97513040651102840211018154660, 7.67717943805798144423115023471, 8.402449026000533424865704733239, 9.627977813884985640099135376941, 10.07024174942785458389292631485, 11.47242509949491631579469741844

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