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Properties

Label 2-504-504.349-c0-0-1
Degree $2$
Conductor $504$
Sign $0.173 - 0.984i$
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  + (−0.5 + 0.866i)2-s + 3-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 0.999·8-s + 9-s − 0.999·10-s + (−0.499 − 0.866i)12-s + (−1 − 1.73i)13-s + (−0.499 − 0.866i)14-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)18-s − 19-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + 3-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 0.999·8-s + 9-s − 0.999·10-s + (−0.499 − 0.866i)12-s + (−1 − 1.73i)13-s + (−0.499 − 0.866i)14-s + (0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)18-s − 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(0.251528\)
Root analytic conductor: \(0.501526\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :0),\ 0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9180494417\)
\(L(\frac12)\) \(\approx\) \(0.9180494417\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 - T \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−10.85615308179037527096033707483, −10.05336237641633738941287371096, −9.548843887652243704977109060494, −8.569730251692255414767893088893, −7.78741213173594247147192461213, −6.90648135148348130200714294975, −6.00478283645380134632712850492, −4.97231901674417063433174229573, −3.24584509188937312741968202860, −2.24204347859469371287362129720, 1.55295682418007375271480264696, 2.67046259978795969709661450435, 4.12076537595443995934849092890, 4.64030483812028349803504844570, 6.73256908674013255759844335816, 7.50429823732128498679165418175, 8.734186844481811898744008869376, 9.135801004569564371320108454935, 9.913468260948056977983502797938, 10.63730981310339958248220375650

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