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Properties

Label 2-120-15.14-c8-0-46
Degree $2$
Conductor $120$
Sign $-0.818 + 0.573i$
Analytic cond. $48.8854$
Root an. cond. $6.99181$
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  + (58.3 − 56.1i)3-s + (617. + 96.4i)5-s − 3.50e3i·7-s + (252. − 6.55e3i)9-s − 2.07e4i·11-s + 2.12e4i·13-s + (4.14e4 − 2.90e4i)15-s − 1.40e5·17-s − 1.82e5·19-s + (−1.97e5 − 2.04e5i)21-s + 4.06e5·23-s + (3.72e5 + 1.19e5i)25-s + (−3.53e5 − 3.96e5i)27-s − 8.58e4i·29-s + 4.69e5·31-s + ⋯
L(s)  = 1  + (0.720 − 0.693i)3-s + (0.988 + 0.154i)5-s − 1.46i·7-s + (0.0385 − 0.999i)9-s − 1.41i·11-s + 0.742i·13-s + (0.818 − 0.573i)15-s − 1.68·17-s − 1.40·19-s + (−1.01 − 1.05i)21-s + 1.45·23-s + (0.952 + 0.304i)25-s + (−0.665 − 0.746i)27-s − 0.121i·29-s + 0.508·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 + 0.573i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.818 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.818 + 0.573i$
Analytic conductor: \(48.8854\)
Root analytic conductor: \(6.99181\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :4),\ -0.818 + 0.573i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.553299530\)
\(L(\frac12)\) \(\approx\) \(2.553299530\)
\(L(5)\) 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 \( 1 + (-58.3 + 56.1i)T \)
5 \( 1 + (-617. - 96.4i)T \)
good7 \( 1 + 3.50e3iT - 5.76e6T^{2} \)
11 \( 1 + 2.07e4iT - 2.14e8T^{2} \)
13 \( 1 - 2.12e4iT - 8.15e8T^{2} \)
17 \( 1 + 1.40e5T + 6.97e9T^{2} \)
19 \( 1 + 1.82e5T + 1.69e10T^{2} \)
23 \( 1 - 4.06e5T + 7.83e10T^{2} \)
29 \( 1 + 8.58e4iT - 5.00e11T^{2} \)
31 \( 1 - 4.69e5T + 8.52e11T^{2} \)
37 \( 1 - 2.30e6iT - 3.51e12T^{2} \)
41 \( 1 - 1.66e6iT - 7.98e12T^{2} \)
43 \( 1 + 1.85e6iT - 1.16e13T^{2} \)
47 \( 1 + 5.76e6T + 2.38e13T^{2} \)
53 \( 1 - 5.45e6T + 6.22e13T^{2} \)
59 \( 1 - 9.28e6iT - 1.46e14T^{2} \)
61 \( 1 - 7.59e6T + 1.91e14T^{2} \)
67 \( 1 + 3.94e7iT - 4.06e14T^{2} \)
71 \( 1 + 1.39e7iT - 6.45e14T^{2} \)
73 \( 1 + 1.62e7iT - 8.06e14T^{2} \)
79 \( 1 - 9.06e6T + 1.51e15T^{2} \)
83 \( 1 + 4.45e7T + 2.25e15T^{2} \)
89 \( 1 + 6.34e6iT - 3.93e15T^{2} \)
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

−11.31303177589675303757828278474, −10.49918214298215391535293803619, −9.170494550281008101581800686363, −8.378335726696595133997658460105, −6.84541346403727465708752283983, −6.41232346826142043029432709817, −4.42950865229477445218341124712, −3.05264859510057372515722583215, −1.74486274913957960004662844823, −0.56483576515663365970127331693, 2.05073867477908988579714132041, 2.58207712736818677250114263657, 4.50508161542762167447440911104, 5.42053594486148681269451645747, 6.79884019567094472074937473138, 8.538503655627768558724358022934, 9.088630985017814915445371592926, 10.02021838032732516555727621119, 11.05628822220202950892946368033, 12.71661958000463470793429945418

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