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Properties

Label 2-120-5.4-c3-0-2
Degree $2$
Conductor $120$
Sign $-0.0160 - 0.999i$
Analytic cond. $7.08022$
Root an. cond. $2.66087$
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  + 3i·3-s + (−0.178 − 11.1i)5-s + 35.0i·7-s − 9·9-s + 25.6·11-s + 37.6i·13-s + (33.5 − 0.536i)15-s + 95.7i·17-s − 50.8·19-s − 105.·21-s + 110. i·23-s + (−124. + 4i)25-s − 27i·27-s + 54.5·29-s + 198.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (−0.0160 − 0.999i)5-s + 1.89i·7-s − 0.333·9-s + 0.702·11-s + 0.803i·13-s + (0.577 − 0.00923i)15-s + 1.36i·17-s − 0.614·19-s − 1.09·21-s + 1.00i·23-s + (−0.999 + 0.0320i)25-s − 0.192i·27-s + 0.349·29-s + 1.14·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0160 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0160 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.0160 - 0.999i$
Analytic conductor: \(7.08022\)
Root analytic conductor: \(2.66087\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :3/2),\ -0.0160 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.00016 + 1.01629i\)
\(L(\frac12)\) \(\approx\) \(1.00016 + 1.01629i\)
\(L(\frac{5}{2})\) 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 \( 1 - 3iT \)
5 \( 1 + (0.178 + 11.1i)T \)
good7 \( 1 - 35.0iT - 343T^{2} \)
11 \( 1 - 25.6T + 1.33e3T^{2} \)
13 \( 1 - 37.6iT - 2.19e3T^{2} \)
17 \( 1 - 95.7iT - 4.91e3T^{2} \)
19 \( 1 + 50.8T + 6.85e3T^{2} \)
23 \( 1 - 110. iT - 1.21e4T^{2} \)
29 \( 1 - 54.5T + 2.43e4T^{2} \)
31 \( 1 - 198.T + 2.97e4T^{2} \)
37 \( 1 + 266. iT - 5.06e4T^{2} \)
41 \( 1 - 103.T + 6.89e4T^{2} \)
43 \( 1 - 108iT - 7.95e4T^{2} \)
47 \( 1 + 597. iT - 1.03e5T^{2} \)
53 \( 1 + 305. iT - 1.48e5T^{2} \)
59 \( 1 - 223.T + 2.05e5T^{2} \)
61 \( 1 - 485.T + 2.26e5T^{2} \)
67 \( 1 - 876. iT - 3.00e5T^{2} \)
71 \( 1 - 585.T + 3.57e5T^{2} \)
73 \( 1 + 1.13e3iT - 3.89e5T^{2} \)
79 \( 1 + 685.T + 4.93e5T^{2} \)
83 \( 1 - 305. iT - 5.71e5T^{2} \)
89 \( 1 + 887.T + 7.04e5T^{2} \)
97 \( 1 - 556. iT - 9.12e5T^{2} \)
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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

−13.05192109501944838396503415050, −12.11807439073390823992758252608, −11.48527701061669095071560294147, −9.831280694064448654710245180772, −8.905456424066426732818790223438, −8.404399285178460973213610555870, −6.24877512724909560348703093536, −5.29910222218419789012278081000, −3.98283864078735772869535659546, −1.98614592785421455403547198093, 0.78487325745986751004655743653, 2.95304287587602149070490726955, 4.38897258711049096191606133481, 6.43602118167436456770529261870, 7.11271283942209307565796333633, 8.075174455443012467585554249264, 9.856823163376597715731293943724, 10.67781277471578965879239303960, 11.56167035025881223762813448658, 12.90145817088641617253109624584

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