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Properties

Label 2-240-12.11-c7-0-49
Degree $2$
Conductor $240$
Sign $-0.757 + 0.652i$
Analytic cond. $74.9724$
Root an. cond. $8.65866$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (15.4 − 44.1i)3-s − 125i·5-s − 3.33i·7-s + (−1.71e3 − 1.36e3i)9-s + 833.·11-s + 1.22e4·13-s + (−5.51e3 − 1.92e3i)15-s + 1.20e4i·17-s − 3.83e4i·19-s + (−147. − 51.4i)21-s + 1.07e5·23-s − 1.56e4·25-s + (−8.65e4 + 5.44e4i)27-s − 2.02e5i·29-s − 3.65e4i·31-s + ⋯
L(s)  = 1  + (0.330 − 0.943i)3-s − 0.447i·5-s − 0.00367i·7-s + (−0.782 − 0.623i)9-s + 0.188·11-s + 1.54·13-s + (−0.422 − 0.147i)15-s + 0.592i·17-s − 1.28i·19-s + (−0.00346 − 0.00121i)21-s + 1.84·23-s − 0.199·25-s + (−0.846 + 0.532i)27-s − 1.54i·29-s − 0.220i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 + 0.652i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.757 + 0.652i$
Analytic conductor: \(74.9724\)
Root analytic conductor: \(8.65866\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :7/2),\ -0.757 + 0.652i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.326273890\)
\(L(\frac12)\) \(\approx\) \(2.326273890\)
\(L(\frac{9}{2})\) 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 \( 1 + (-15.4 + 44.1i)T \)
5 \( 1 + 125iT \)
good7 \( 1 + 3.33iT - 8.23e5T^{2} \)
11 \( 1 - 833.T + 1.94e7T^{2} \)
13 \( 1 - 1.22e4T + 6.27e7T^{2} \)
17 \( 1 - 1.20e4iT - 4.10e8T^{2} \)
19 \( 1 + 3.83e4iT - 8.93e8T^{2} \)
23 \( 1 - 1.07e5T + 3.40e9T^{2} \)
29 \( 1 + 2.02e5iT - 1.72e10T^{2} \)
31 \( 1 + 3.65e4iT - 2.75e10T^{2} \)
37 \( 1 + 3.38e5T + 9.49e10T^{2} \)
41 \( 1 + 3.13e5iT - 1.94e11T^{2} \)
43 \( 1 + 6.19e4iT - 2.71e11T^{2} \)
47 \( 1 - 9.58e5T + 5.06e11T^{2} \)
53 \( 1 - 2.57e5iT - 1.17e12T^{2} \)
59 \( 1 + 1.95e6T + 2.48e12T^{2} \)
61 \( 1 + 1.08e6T + 3.14e12T^{2} \)
67 \( 1 - 8.33e5iT - 6.06e12T^{2} \)
71 \( 1 + 2.51e6T + 9.09e12T^{2} \)
73 \( 1 + 2.38e6T + 1.10e13T^{2} \)
79 \( 1 - 4.63e6iT - 1.92e13T^{2} \)
83 \( 1 + 1.20e6T + 2.71e13T^{2} \)
89 \( 1 + 3.27e6iT - 4.42e13T^{2} \)
97 \( 1 - 1.27e7T + 8.07e13T^{2} \)
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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

−10.71427033685788067353754592978, −9.044064026343321003920515265233, −8.712806248453018176396693516113, −7.51559944554271277049552910963, −6.54378513155396122767861945546, −5.57925104285012608957300313987, −4.06485814891392858466579122261, −2.80703069304025603821589794600, −1.46070844261841757809377837132, −0.56137171839748330568801238138, 1.34301237768354099566999306748, 3.00651841134182930881314840640, 3.73231160853729408418532894945, 5.02322351022455597385254714139, 6.09751704092242177844137859060, 7.34705249827500725934720844480, 8.619562568764268986068576919528, 9.195785715667921476670907343591, 10.51667015433800246233103177817, 10.87653922913998762757779473541

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