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Properties

Label 2-240-15.14-c8-0-15
Degree $2$
Conductor $240$
Sign $-0.432 - 0.901i$
Analytic cond. $97.7708$
Root an. cond. $9.88791$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (74.7 − 31.2i)3-s + (31.8 + 624. i)5-s − 3.42e3i·7-s + (4.61e3 − 4.66e3i)9-s + 1.57e4i·11-s + 2.70e4i·13-s + (2.18e4 + 4.56e4i)15-s + 1.42e4·17-s − 2.10e5·19-s + (−1.07e5 − 2.56e5i)21-s − 6.02e4·23-s + (−3.88e5 + 3.97e4i)25-s + (1.98e5 − 4.92e5i)27-s + 7.58e5i·29-s − 3.33e5·31-s + ⋯
L(s)  = 1  + (0.922 − 0.385i)3-s + (0.0509 + 0.998i)5-s − 1.42i·7-s + (0.702 − 0.711i)9-s + 1.07i·11-s + 0.947i·13-s + (0.432 + 0.901i)15-s + 0.170·17-s − 1.61·19-s + (−0.550 − 1.31i)21-s − 0.215·23-s + (−0.994 + 0.101i)25-s + (0.374 − 0.927i)27-s + 1.07i·29-s − 0.360·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.432 - 0.901i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.432 - 0.901i$
Analytic conductor: \(97.7708\)
Root analytic conductor: \(9.88791\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :4),\ -0.432 - 0.901i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.590875211\)
\(L(\frac12)\) \(\approx\) \(1.590875211\)
\(L(5)\) 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 \( 1 + (-74.7 + 31.2i)T \)
5 \( 1 + (-31.8 - 624. i)T \)
good7 \( 1 + 3.42e3iT - 5.76e6T^{2} \)
11 \( 1 - 1.57e4iT - 2.14e8T^{2} \)
13 \( 1 - 2.70e4iT - 8.15e8T^{2} \)
17 \( 1 - 1.42e4T + 6.97e9T^{2} \)
19 \( 1 + 2.10e5T + 1.69e10T^{2} \)
23 \( 1 + 6.02e4T + 7.83e10T^{2} \)
29 \( 1 - 7.58e5iT - 5.00e11T^{2} \)
31 \( 1 + 3.33e5T + 8.52e11T^{2} \)
37 \( 1 - 6.83e5iT - 3.51e12T^{2} \)
41 \( 1 - 1.23e6iT - 7.98e12T^{2} \)
43 \( 1 + 1.15e6iT - 1.16e13T^{2} \)
47 \( 1 + 1.52e6T + 2.38e13T^{2} \)
53 \( 1 - 4.73e6T + 6.22e13T^{2} \)
59 \( 1 - 1.30e7iT - 1.46e14T^{2} \)
61 \( 1 + 6.62e6T + 1.91e14T^{2} \)
67 \( 1 - 2.47e7iT - 4.06e14T^{2} \)
71 \( 1 + 1.07e7iT - 6.45e14T^{2} \)
73 \( 1 + 3.48e7iT - 8.06e14T^{2} \)
79 \( 1 + 5.01e7T + 1.51e15T^{2} \)
83 \( 1 - 4.50e7T + 2.25e15T^{2} \)
89 \( 1 - 7.30e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.45e8iT - 7.83e15T^{2} \)
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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

−10.70859044315547322057843283919, −10.13705416406440820782646546208, −9.097929693105060923336605585958, −7.83092931897007655197487533432, −7.03620988505622181660757066333, −6.55524762458778547317817103525, −4.40346130267760718342126542450, −3.69922503960582176876749607083, −2.37457271402539396405068565321, −1.43102902648987674610598613268, 0.27461262407561322581255198549, 1.85309100384166495406117539670, 2.82490268867028992906520747075, 4.05226388573315281354934701107, 5.26350933537399571322268000269, 6.06860593464950920703953892847, 7.968005342906942630347239933234, 8.526655962811455338984300670928, 9.128161932422761506519653833237, 10.16852934885799186980877225723

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