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Properties

Label 2-120-15.14-c8-0-47
Degree $2$
Conductor $120$
Sign $-0.967 - 0.251i$
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  + (48.4 − 64.9i)3-s + (−236. − 578. i)5-s − 3.30e3i·7-s + (−1.86e3 − 6.29e3i)9-s − 1.11e4i·11-s − 1.97e4i·13-s + (−4.90e4 − 1.27e4i)15-s + 9.77e3·17-s + 1.39e5·19-s + (−2.14e5 − 1.60e5i)21-s − 6.41e4·23-s + (−2.79e5 + 2.73e5i)25-s + (−4.98e5 − 1.83e5i)27-s + 3.41e4i·29-s + 1.18e6·31-s + ⋯
L(s)  = 1  + (0.598 − 0.801i)3-s + (−0.377 − 0.925i)5-s − 1.37i·7-s + (−0.284 − 0.958i)9-s − 0.758i·11-s − 0.690i·13-s + (−0.967 − 0.251i)15-s + 0.117·17-s + 1.07·19-s + (−1.10 − 0.824i)21-s − 0.229·23-s + (−0.714 + 0.699i)25-s + (−0.938 − 0.346i)27-s + 0.0482i·29-s + 1.28·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.251i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.967 - 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.967 - 0.251i$
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: \((2,\ 120,\ (\ :4),\ -0.967 - 0.251i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.082764605\)
\(L(\frac12)\) \(\approx\) \(2.082764605\)
\(L(5)\) 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 \( 1 + (-48.4 + 64.9i)T \)
5 \( 1 + (236. + 578. i)T \)
good7 \( 1 + 3.30e3iT - 5.76e6T^{2} \)
11 \( 1 + 1.11e4iT - 2.14e8T^{2} \)
13 \( 1 + 1.97e4iT - 8.15e8T^{2} \)
17 \( 1 - 9.77e3T + 6.97e9T^{2} \)
19 \( 1 - 1.39e5T + 1.69e10T^{2} \)
23 \( 1 + 6.41e4T + 7.83e10T^{2} \)
29 \( 1 - 3.41e4iT - 5.00e11T^{2} \)
31 \( 1 - 1.18e6T + 8.52e11T^{2} \)
37 \( 1 + 1.18e6iT - 3.51e12T^{2} \)
41 \( 1 - 2.36e6iT - 7.98e12T^{2} \)
43 \( 1 - 1.21e5iT - 1.16e13T^{2} \)
47 \( 1 - 8.81e6T + 2.38e13T^{2} \)
53 \( 1 - 4.86e6T + 6.22e13T^{2} \)
59 \( 1 - 2.07e7iT - 1.46e14T^{2} \)
61 \( 1 + 1.65e7T + 1.91e14T^{2} \)
67 \( 1 + 1.43e7iT - 4.06e14T^{2} \)
71 \( 1 - 1.59e7iT - 6.45e14T^{2} \)
73 \( 1 + 3.85e7iT - 8.06e14T^{2} \)
79 \( 1 + 6.80e7T + 1.51e15T^{2} \)
83 \( 1 + 2.71e7T + 2.25e15T^{2} \)
89 \( 1 - 3.70e6iT - 3.93e15T^{2} \)
97 \( 1 - 7.17e7iT - 7.83e15T^{2} \)
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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

−11.55965332352390806388648692422, −10.23672958205565274387848312644, −8.984243241715593586372308486725, −7.957475123919870185386879707070, −7.30462823063791187673530627302, −5.80758515748035129739969536177, −4.22617981792744090792613341940, −3.09713095648252942556937031694, −1.17106099413400853402231456637, −0.57433204005836044042916379390, 2.15537197013628865912767151378, 3.06427074206857159812239724029, 4.39087673457419623840239991595, 5.70571866730609731603912920151, 7.15803452526275103835613825927, 8.339433475263854694067160808350, 9.383337489234484352659005112108, 10.20972334820837720028629002833, 11.46883755568302865044561014863, 12.19149784851514777246115051526

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