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Properties

Label 2-360-5.4-c5-0-17
Degree $2$
Conductor $360$
Sign $0.907 + 0.419i$
Analytic cond. $57.7381$
Root an. cond. $7.59856$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−23.4 + 50.7i)5-s − 10.2i·7-s − 596.·11-s − 420. i·13-s + 974. i·17-s + 380.·19-s − 3.54e3i·23-s + (−2.02e3 − 2.37e3i)25-s + 5.44e3·29-s − 3.62e3·31-s + (520. + 240. i)35-s − 1.75e3i·37-s − 263.·41-s + 1.44e4i·43-s + 2.34e4i·47-s + ⋯
L(s)  = 1  + (−0.419 + 0.907i)5-s − 0.0791i·7-s − 1.48·11-s − 0.690i·13-s + 0.817i·17-s + 0.241·19-s − 1.39i·23-s + (−0.648 − 0.760i)25-s + 1.20·29-s − 0.677·31-s + (0.0718 + 0.0331i)35-s − 0.210i·37-s − 0.0245·41-s + 1.18i·43-s + 1.54i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.907 + 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.907 + 0.419i$
Analytic conductor: \(57.7381\)
Root analytic conductor: \(7.59856\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :5/2),\ 0.907 + 0.419i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.294081119\)
\(L(\frac12)\) \(\approx\) \(1.294081119\)
\(L(\frac{7}{2})\) 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 \( 1 \)
5 \( 1 + (23.4 - 50.7i)T \)
good7 \( 1 + 10.2iT - 1.68e4T^{2} \)
11 \( 1 + 596.T + 1.61e5T^{2} \)
13 \( 1 + 420. iT - 3.71e5T^{2} \)
17 \( 1 - 974. iT - 1.41e6T^{2} \)
19 \( 1 - 380.T + 2.47e6T^{2} \)
23 \( 1 + 3.54e3iT - 6.43e6T^{2} \)
29 \( 1 - 5.44e3T + 2.05e7T^{2} \)
31 \( 1 + 3.62e3T + 2.86e7T^{2} \)
37 \( 1 + 1.75e3iT - 6.93e7T^{2} \)
41 \( 1 + 263.T + 1.15e8T^{2} \)
43 \( 1 - 1.44e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.34e4iT - 2.29e8T^{2} \)
53 \( 1 + 3.34e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.90e3T + 7.14e8T^{2} \)
61 \( 1 - 2.94e4T + 8.44e8T^{2} \)
67 \( 1 + 7.16e3iT - 1.35e9T^{2} \)
71 \( 1 - 8.13e4T + 1.80e9T^{2} \)
73 \( 1 - 5.51e4iT - 2.07e9T^{2} \)
79 \( 1 + 1.64e4T + 3.07e9T^{2} \)
83 \( 1 + 1.16e5iT - 3.93e9T^{2} \)
89 \( 1 - 9.93e4T + 5.58e9T^{2} \)
97 \( 1 + 6.29e4iT - 8.58e9T^{2} \)
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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

−10.56158659797824667882474667972, −9.971940795003745027417669906404, −8.390462140693256753834677434208, −7.83101563181010176479898599577, −6.78293604292268960663179432652, −5.76177459273596540927742797662, −4.56077745650809606065535494678, −3.24693795440235347927205025992, −2.37555874426123544986831821737, −0.46232799497965264171112083621, 0.77819178037057338296697472791, 2.25601706309143325861686925265, 3.64884735804864645189521455610, 4.91301569957609372450127674247, 5.52141457512213568404583117522, 7.09104443659536124623559476172, 7.88287356198833774669121496592, 8.819629551302369354828767062672, 9.658723196631004880265159087487, 10.70568362580650018749145414225

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