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Properties

Label 2-504-56.5-c0-0-0
Degree $2$
Conductor $504$
Sign $0.126 - 0.991i$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.866 + 1.5i)5-s + (−0.5 − 0.866i)7-s + 0.999i·8-s + (−1.5 + 0.866i)10-s + (0.866 − 0.5i)11-s − 0.999i·14-s + (−0.5 + 0.866i)16-s − 1.73·20-s + 0.999·22-s + (−1 − 1.73i)25-s + (0.499 − 0.866i)28-s − i·29-s + (1.5 − 0.866i)31-s + (−0.866 + 0.499i)32-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.866 + 1.5i)5-s + (−0.5 − 0.866i)7-s + 0.999i·8-s + (−1.5 + 0.866i)10-s + (0.866 − 0.5i)11-s − 0.999i·14-s + (−0.5 + 0.866i)16-s − 1.73·20-s + 0.999·22-s + (−1 − 1.73i)25-s + (0.499 − 0.866i)28-s − i·29-s + (1.5 − 0.866i)31-s + (−0.866 + 0.499i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.126 - 0.991i$
Analytic conductor: \(0.251528\)
Root analytic conductor: \(0.501526\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :0),\ 0.126 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.216440773\)
\(L(\frac12)\) \(\approx\) \(1.216440773\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + 1.73T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - 1.73iT - T^{2} \)
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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

−11.40477886991716214683257508917, −10.77976909885173084239057698408, −9.723632715718002728090957826206, −8.192571633529719359469135524956, −7.49308838378267155463450931873, −6.62160194678349031620058971862, −6.16867248972288551448118932948, −4.36946494951643178729483467631, −3.66621118333488728812762425350, −2.79665172562325305802621206087, 1.45716744033243980861419320845, 3.14465577786259674417037963900, 4.30278264221144197083951819993, 4.98533711308686262841477522257, 6.03520618585606036084204369645, 7.12704391063918835006708472963, 8.495418204977035667327019380250, 9.156351174177907202121396511722, 10.04070669176029521249205264966, 11.36954098805936403509236898391

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