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Properties

Label 2-120-5.4-c3-0-4
Degree $2$
Conductor $120$
Sign $0.999 + 0.0160i$
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 + (11.1 + 0.178i)5-s − 33.0i·7-s − 9·9-s + 48.3·11-s + 60.3i·13-s + (−0.536 + 33.5i)15-s − 17.7i·17-s + 130.·19-s + 99.2·21-s − 70.8i·23-s + (124. + 4i)25-s − 27i·27-s − 104.·29-s − 210.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.999 + 0.0160i)5-s − 1.78i·7-s − 0.333·9-s + 1.32·11-s + 1.28i·13-s + (−0.00923 + 0.577i)15-s − 0.253i·17-s + 1.58·19-s + 1.03·21-s − 0.642i·23-s + (0.999 + 0.0320i)25-s − 0.192i·27-s − 0.669·29-s − 1.21·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.999 + 0.0160i$
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: \((2,\ 120,\ (\ :3/2),\ 0.999 + 0.0160i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.97115 - 0.0157722i\)
\(L(\frac12)\) \(\approx\) \(1.97115 - 0.0157722i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-11.1 - 0.178i)T \)
good7 \( 1 + 33.0iT - 343T^{2} \)
11 \( 1 - 48.3T + 1.33e3T^{2} \)
13 \( 1 - 60.3iT - 2.19e3T^{2} \)
17 \( 1 + 17.7iT - 4.91e3T^{2} \)
19 \( 1 - 130.T + 6.85e3T^{2} \)
23 \( 1 + 70.8iT - 1.21e4T^{2} \)
29 \( 1 + 104.T + 2.43e4T^{2} \)
31 \( 1 + 210.T + 2.97e4T^{2} \)
37 \( 1 - 300. iT - 5.06e4T^{2} \)
41 \( 1 - 240.T + 6.89e4T^{2} \)
43 \( 1 - 108iT - 7.95e4T^{2} \)
47 \( 1 + 278. iT - 1.03e5T^{2} \)
53 \( 1 + 328. iT - 1.48e5T^{2} \)
59 \( 1 + 889.T + 2.05e5T^{2} \)
61 \( 1 + 241.T + 2.26e5T^{2} \)
67 \( 1 - 103. iT - 3.00e5T^{2} \)
71 \( 1 + 277.T + 3.57e5T^{2} \)
73 \( 1 + 274. iT - 3.89e5T^{2} \)
79 \( 1 + 366.T + 4.93e5T^{2} \)
83 \( 1 + 57.7iT - 5.71e5T^{2} \)
89 \( 1 - 203.T + 7.04e5T^{2} \)
97 \( 1 - 1.28e3iT - 9.12e5T^{2} \)
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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

−13.35889354695434387026004174657, −11.75283543579525608574561280666, −10.80316602975846990912315281459, −9.740022623030560319470314271253, −9.164387088752878325423512724974, −7.31455869948494938565759026149, −6.39164619785762993858077139790, −4.74550838166045550077459762798, −3.62980877620348759504541606460, −1.34871222697443959106075745453, 1.59749681186365131846113244135, 3.01376935274297240641932811492, 5.53345386431883575173458458270, 5.94932090656668889734369235222, 7.52160920096974116156534185047, 8.999138419914933678607168032711, 9.459691099489557829666822620245, 11.10597781208159789355578545281, 12.20398311071718538353629575128, 12.81681192304772813016565827899

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