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Properties

Label 2-504-21.17-c5-0-0
Degree $2$
Conductor $504$
Sign $-0.988 - 0.154i$
Analytic cond. $80.8334$
Root an. cond. $8.99074$
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  + (−14.9 − 25.9i)5-s + (−128. + 19.3i)7-s + (321. + 185. i)11-s + 148. i·13-s + (−717. + 1.24e3i)17-s + (2.41e3 − 1.39e3i)19-s + (232. − 134. i)23-s + (1.11e3 − 1.92e3i)25-s − 7.53e3i·29-s + (−2.68e3 − 1.54e3i)31-s + (2.42e3 + 3.03e3i)35-s + (7.52e3 + 1.30e4i)37-s + 2.24e3·41-s − 1.77e4·43-s + (−9.99e3 − 1.73e4i)47-s + ⋯
L(s)  = 1  + (−0.268 − 0.464i)5-s + (−0.988 + 0.149i)7-s + (0.801 + 0.462i)11-s + 0.242i·13-s + (−0.602 + 1.04i)17-s + (1.53 − 0.885i)19-s + (0.0915 − 0.0528i)23-s + (0.356 − 0.616i)25-s − 1.66i·29-s + (−0.501 − 0.289i)31-s + (0.334 + 0.419i)35-s + (0.903 + 1.56i)37-s + 0.208·41-s − 1.46·43-s + (−0.660 − 1.14i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.154i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.154i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.988 - 0.154i$
Analytic conductor: \(80.8334\)
Root analytic conductor: \(8.99074\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :5/2),\ -0.988 - 0.154i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.01939130198\)
\(L(\frac12)\) \(\approx\) \(0.01939130198\)
\(L(\frac{7}{2})\) 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 \( 1 \)
7 \( 1 + (128. - 19.3i)T \)
good5 \( 1 + (14.9 + 25.9i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-321. - 185. i)T + (8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 148. iT - 3.71e5T^{2} \)
17 \( 1 + (717. - 1.24e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-2.41e3 + 1.39e3i)T + (1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-232. + 134. i)T + (3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 7.53e3iT - 2.05e7T^{2} \)
31 \( 1 + (2.68e3 + 1.54e3i)T + (1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-7.52e3 - 1.30e4i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 2.24e3T + 1.15e8T^{2} \)
43 \( 1 + 1.77e4T + 1.47e8T^{2} \)
47 \( 1 + (9.99e3 + 1.73e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (4.17e3 + 2.40e3i)T + (2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (2.08e4 - 3.61e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (4.30e4 - 2.48e4i)T + (4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (2.52e4 - 4.38e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 2.94e4iT - 1.80e9T^{2} \)
73 \( 1 + (4.03e4 + 2.33e4i)T + (1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-7.86e3 - 1.36e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 9.38e4T + 3.93e9T^{2} \)
89 \( 1 + (-1.69e4 - 2.93e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.15e5iT - 8.58e9T^{2} \)
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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

−10.39551494818781500870611552598, −9.547019210084181299553488264725, −8.936906062705961635045349232330, −7.88177575277838768269912843726, −6.78076241188385140854510509946, −6.10635605120725182661220667214, −4.77472446551225498812811337124, −3.88589534336790792065583788985, −2.71091035987673261246365360785, −1.26982414750708762750147437117, 0.00482680463135114437356507200, 1.31906061503688877521131993066, 3.08260436202328119683626019135, 3.52275811989811140744963754714, 5.00475452996236701747361588482, 6.11365840285188949359422831268, 6.98476815170403249807803904953, 7.68568456681266533743675899748, 9.125626380388914590255641228635, 9.499443216265934881263569422105

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