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Properties

Label 2-120-120.59-c3-0-10
Degree $2$
Conductor $120$
Sign $-0.916 + 0.399i$
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  + (−2.12 + 1.86i)2-s + (0.659 + 5.15i)3-s + (1.02 − 7.93i)4-s + (10.6 + 3.40i)5-s + (−11.0 − 9.71i)6-s − 28.2·7-s + (12.6 + 18.7i)8-s + (−26.1 + 6.79i)9-s + (−28.9 + 12.6i)10-s + 38.5i·11-s + (41.5 + 0.0568i)12-s + 36.1·13-s + (60.0 − 52.7i)14-s + (−10.5 + 57.1i)15-s + (−61.8 − 16.2i)16-s − 74.9·17-s + ⋯
L(s)  = 1  + (−0.751 + 0.660i)2-s + (0.126 + 0.991i)3-s + (0.128 − 0.991i)4-s + (0.952 + 0.304i)5-s + (−0.750 − 0.661i)6-s − 1.52·7-s + (0.558 + 0.829i)8-s + (−0.967 + 0.251i)9-s + (−0.916 + 0.400i)10-s + 1.05i·11-s + (0.999 + 0.00136i)12-s + 0.772·13-s + (1.14 − 1.00i)14-s + (−0.180 + 0.983i)15-s + (−0.967 − 0.254i)16-s − 1.06·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.916 + 0.399i$
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: \((2,\ 120,\ (\ :3/2),\ -0.916 + 0.399i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.131671 - 0.632380i\)
\(L(\frac12)\) \(\approx\) \(0.131671 - 0.632380i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.12 - 1.86i)T \)
3 \( 1 + (-0.659 - 5.15i)T \)
5 \( 1 + (-10.6 - 3.40i)T \)
good7 \( 1 + 28.2T + 343T^{2} \)
11 \( 1 - 38.5iT - 1.33e3T^{2} \)
13 \( 1 - 36.1T + 2.19e3T^{2} \)
17 \( 1 + 74.9T + 4.91e3T^{2} \)
19 \( 1 + 136.T + 6.85e3T^{2} \)
23 \( 1 + 114. iT - 1.21e4T^{2} \)
29 \( 1 - 109.T + 2.43e4T^{2} \)
31 \( 1 - 57.1iT - 2.97e4T^{2} \)
37 \( 1 + 100.T + 5.06e4T^{2} \)
41 \( 1 - 173. iT - 6.89e4T^{2} \)
43 \( 1 - 86.0iT - 7.95e4T^{2} \)
47 \( 1 - 239. iT - 1.03e5T^{2} \)
53 \( 1 - 476. iT - 1.48e5T^{2} \)
59 \( 1 - 762. iT - 2.05e5T^{2} \)
61 \( 1 + 614. iT - 2.26e5T^{2} \)
67 \( 1 - 382. iT - 3.00e5T^{2} \)
71 \( 1 - 124.T + 3.57e5T^{2} \)
73 \( 1 + 267. iT - 3.89e5T^{2} \)
79 \( 1 - 586. iT - 4.93e5T^{2} \)
83 \( 1 - 282.T + 5.71e5T^{2} \)
89 \( 1 - 604. iT - 7.04e5T^{2} \)
97 \( 1 - 1.04e3iT - 9.12e5T^{2} \)
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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

−13.81924295120473536523271192532, −12.77724708226569472123106738543, −10.78778758088421143630319141809, −10.24801366798614294425690673004, −9.368029222576211383134809561871, −8.645828354749741486238602958082, −6.67882061837249418587286764305, −6.13794946534348612098693464191, −4.50794006113644358474796734920, −2.49609643084857185970011568461, 0.39898368801665136970311037706, 2.09516696493404696789250174841, 3.42578455494788371976579617421, 6.10250121045089211593084364411, 6.76159578621633473742073865639, 8.513759897983760355415093670534, 9.033414444035140527576951093391, 10.27103295859436123264563734417, 11.32844545252836352191000080350, 12.64847753048013864472904258903

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