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Properties

Label 2-120-120.59-c3-0-4
Degree $2$
Conductor $120$
Sign $-0.793 - 0.609i$
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  + (−2.80 + 0.363i)2-s + (3.73 − 3.61i)3-s + (7.73 − 2.03i)4-s + (−2.66 + 10.8i)5-s + (−9.15 + 11.4i)6-s − 26.3·7-s + (−20.9 + 8.53i)8-s + (0.853 − 26.9i)9-s + (3.52 − 31.4i)10-s + 37.4i·11-s + (21.4 − 35.5i)12-s − 30.8·13-s + (73.8 − 9.57i)14-s + (29.3 + 50.1i)15-s + (55.6 − 31.5i)16-s − 54.0·17-s + ⋯
L(s)  = 1  + (−0.991 + 0.128i)2-s + (0.718 − 0.695i)3-s + (0.966 − 0.254i)4-s + (−0.238 + 0.971i)5-s + (−0.622 + 0.782i)6-s − 1.42·7-s + (−0.926 + 0.377i)8-s + (0.0315 − 0.999i)9-s + (0.111 − 0.993i)10-s + 1.02i·11-s + (0.517 − 0.855i)12-s − 0.658·13-s + (1.41 − 0.182i)14-s + (0.504 + 0.863i)15-s + (0.869 − 0.493i)16-s − 0.770·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 - 0.609i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.793 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.793 - 0.609i$
Analytic conductor: \(7.08022\)
Root analytic conductor: \(2.66087\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :3/2),\ -0.793 - 0.609i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.107287 + 0.315801i\)
\(L(\frac12)\) \(\approx\) \(0.107287 + 0.315801i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.80 - 0.363i)T \)
3 \( 1 + (-3.73 + 3.61i)T \)
5 \( 1 + (2.66 - 10.8i)T \)
good7 \( 1 + 26.3T + 343T^{2} \)
11 \( 1 - 37.4iT - 1.33e3T^{2} \)
13 \( 1 + 30.8T + 2.19e3T^{2} \)
17 \( 1 + 54.0T + 4.91e3T^{2} \)
19 \( 1 + 4.70T + 6.85e3T^{2} \)
23 \( 1 - 129. iT - 1.21e4T^{2} \)
29 \( 1 + 230.T + 2.43e4T^{2} \)
31 \( 1 - 123. iT - 2.97e4T^{2} \)
37 \( 1 - 349.T + 5.06e4T^{2} \)
41 \( 1 - 74.8iT - 6.89e4T^{2} \)
43 \( 1 - 364. iT - 7.95e4T^{2} \)
47 \( 1 + 45.7iT - 1.03e5T^{2} \)
53 \( 1 + 682. iT - 1.48e5T^{2} \)
59 \( 1 - 256. iT - 2.05e5T^{2} \)
61 \( 1 + 435. iT - 2.26e5T^{2} \)
67 \( 1 + 862. iT - 3.00e5T^{2} \)
71 \( 1 + 366.T + 3.57e5T^{2} \)
73 \( 1 + 215. iT - 3.89e5T^{2} \)
79 \( 1 - 340. iT - 4.93e5T^{2} \)
83 \( 1 - 605.T + 5.71e5T^{2} \)
89 \( 1 - 517. iT - 7.04e5T^{2} \)
97 \( 1 - 1.44e3iT - 9.12e5T^{2} \)
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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

−13.30829355614916009717298802764, −12.36618863162219233389018709150, −11.22997690064226830232576704341, −9.781303622328604896882767054128, −9.428988937544831428250940994298, −7.79639071143960963788572039936, −7.04476492639171968270027678413, −6.30333512571246476411855801100, −3.35949440418777668274287292707, −2.19734199452398271760378449969, 0.20185989247542140925974150557, 2.62162550851288171926658322204, 3.99395356469170939105476339018, 5.91999431233180023122357052053, 7.47738882025109264166447331254, 8.694304580940633455214773014814, 9.239323801054179372515518876786, 10.14704323953590185144443593507, 11.28737798667626105595838983620, 12.62516453249140173841957261453

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