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Properties

Label 2-504-504.349-c0-0-3
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 + i·3-s + (−0.499 − 0.866i)4-s + (−0.866 − 1.5i)5-s + (0.866 + 0.5i)6-s + (0.5 − 0.866i)7-s − 0.999·8-s − 9-s − 1.73·10-s + (0.866 − 0.499i)12-s + (−0.499 − 0.866i)14-s + (1.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)18-s + 1.73·19-s + (−0.866 + 1.49i)20-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + i·3-s + (−0.499 − 0.866i)4-s + (−0.866 − 1.5i)5-s + (0.866 + 0.5i)6-s + (0.5 − 0.866i)7-s − 0.999·8-s − 9-s − 1.73·10-s + (0.866 − 0.499i)12-s + (−0.499 − 0.866i)14-s + (1.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)18-s + 1.73·19-s + (−0.866 + 1.49i)20-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

\(L(\frac{1}{2})\) \(\approx\) \(0.9248003092\)
\(L(\frac12)\) \(\approx\) \(0.9248003092\)
\(L(1)\) 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 \( 1 - iT \)
7 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 1.73T + T^{2} \)
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 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T^{2} \)
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

−11.14989902992705414492875802802, −10.02659088933518166318991761433, −9.319589536819593221049397456668, −8.514671110352464865938652911967, −7.50427033555610812423659610660, −5.53439111412024353863594650571, −4.90783215116222407597256070595, −4.15029549931977262429035031629, −3.33083573454878803575661571539, −1.12822473046459807698251101130, 2.61924620992550079111612598540, 3.42048395291419444541520483381, 5.02494698394596078496690802273, 6.09381690610835088742178254572, 6.87213240171755611725731495009, 7.61643387063820742629256065344, 8.185746193896863876157235988995, 9.276253700110870653515629246154, 10.86301612824383687972804432646, 11.71999586499176044976990895085

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