Properties

Label 2-504-7.2-c3-0-14
Degree $2$
Conductor $504$
Sign $0.831 + 0.555i$
Analytic cond. $29.7369$
Root an. cond. $5.45316$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.36 + 5.83i)5-s + (−11.2 + 14.7i)7-s + (−26.8 − 46.4i)11-s + 1.73·13-s + (−23.4 − 40.6i)17-s + (10.0 − 17.4i)19-s + (59.4 − 103. i)23-s + (39.8 + 68.9i)25-s + 103.·29-s + (78.7 + 136. i)31-s + (−48.0 − 115. i)35-s + (18.8 − 32.6i)37-s + 287.·41-s + 504.·43-s + (−110. + 190. i)47-s + ⋯
L(s)  = 1  + (−0.301 + 0.521i)5-s + (−0.606 + 0.794i)7-s + (−0.735 − 1.27i)11-s + 0.0370·13-s + (−0.334 − 0.580i)17-s + (0.121 − 0.210i)19-s + (0.539 − 0.933i)23-s + (0.318 + 0.551i)25-s + 0.663·29-s + (0.456 + 0.790i)31-s + (−0.232 − 0.556i)35-s + (0.0838 − 0.145i)37-s + 1.09·41-s + 1.79·43-s + (−0.342 + 0.592i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.831 + 0.555i$
Analytic conductor: \(29.7369\)
Root analytic conductor: \(5.45316\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :3/2),\ 0.831 + 0.555i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.331667099\)
\(L(\frac12)\) \(\approx\) \(1.331667099\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (11.2 - 14.7i)T \)
good5 \( 1 + (3.36 - 5.83i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (26.8 + 46.4i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 1.73T + 2.19e3T^{2} \)
17 \( 1 + (23.4 + 40.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-10.0 + 17.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-59.4 + 103. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 103.T + 2.43e4T^{2} \)
31 \( 1 + (-78.7 - 136. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-18.8 + 32.6i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 287.T + 6.89e4T^{2} \)
43 \( 1 - 504.T + 7.95e4T^{2} \)
47 \( 1 + (110. - 190. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-146. - 253. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (297. + 516. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-132. + 229. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (468. + 811. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 545.T + 3.57e5T^{2} \)
73 \( 1 + (149. + 259. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-470. + 814. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 611.T + 5.71e5T^{2} \)
89 \( 1 + (588. - 1.01e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.48e3T + 9.12e5T^{2} \)
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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

−10.69783590675558426619184921408, −9.399179531290875639743785655827, −8.719791410761500828768695289473, −7.74095912023939143082649973880, −6.65869333546370496845752594104, −5.86445017541612854891432016055, −4.78025627880001454478427534093, −3.22325738652837513973249848797, −2.64264843154482831505231357713, −0.53631602909245864778121442302, 0.936868229129766546578690048265, 2.52589174722868294605492483958, 3.97369608578588513608872761678, 4.70850881330329908777157530769, 5.95972752126299226092516862454, 7.12168562849464462171296269616, 7.74556752150728592453067581795, 8.835933855188015080243330715947, 9.850282030518498551530573613229, 10.40854172488409887999603960938

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