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Properties

Label 2-120-15.14-c8-0-44
Degree $2$
Conductor $120$
Sign $-0.737 + 0.675i$
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  + (63.1 + 50.7i)3-s + (94.4 − 617. i)5-s − 2.12e3i·7-s + (1.41e3 + 6.40e3i)9-s − 1.88e4i·11-s + 1.61e3i·13-s + (3.73e4 − 3.42e4i)15-s − 1.28e5·17-s + 1.23e5·19-s + (1.07e5 − 1.33e5i)21-s − 4.63e5·23-s + (−3.72e5 − 1.16e5i)25-s + (−2.35e5 + 4.76e5i)27-s + 8.67e5i·29-s − 1.29e6·31-s + ⋯
L(s)  = 1  + (0.779 + 0.626i)3-s + (0.151 − 0.988i)5-s − 0.883i·7-s + (0.215 + 0.976i)9-s − 1.28i·11-s + 0.0564i·13-s + (0.737 − 0.675i)15-s − 1.53·17-s + 0.943·19-s + (0.553 − 0.688i)21-s − 1.65·23-s + (−0.954 − 0.298i)25-s + (−0.443 + 0.896i)27-s + 1.22i·29-s − 1.39·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.737 + 0.675i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.737 + 0.675i$
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: \((2,\ 120,\ (\ :4),\ -0.737 + 0.675i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.361164067\)
\(L(\frac12)\) \(\approx\) \(1.361164067\)
\(L(5)\) 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 \( 1 + (-63.1 - 50.7i)T \)
5 \( 1 + (-94.4 + 617. i)T \)
good7 \( 1 + 2.12e3iT - 5.76e6T^{2} \)
11 \( 1 + 1.88e4iT - 2.14e8T^{2} \)
13 \( 1 - 1.61e3iT - 8.15e8T^{2} \)
17 \( 1 + 1.28e5T + 6.97e9T^{2} \)
19 \( 1 - 1.23e5T + 1.69e10T^{2} \)
23 \( 1 + 4.63e5T + 7.83e10T^{2} \)
29 \( 1 - 8.67e5iT - 5.00e11T^{2} \)
31 \( 1 + 1.29e6T + 8.52e11T^{2} \)
37 \( 1 - 1.60e5iT - 3.51e12T^{2} \)
41 \( 1 + 4.10e6iT - 7.98e12T^{2} \)
43 \( 1 - 1.57e6iT - 1.16e13T^{2} \)
47 \( 1 - 5.70e6T + 2.38e13T^{2} \)
53 \( 1 + 8.30e6T + 6.22e13T^{2} \)
59 \( 1 + 1.28e7iT - 1.46e14T^{2} \)
61 \( 1 - 2.79e6T + 1.91e14T^{2} \)
67 \( 1 - 2.02e7iT - 4.06e14T^{2} \)
71 \( 1 + 2.19e7iT - 6.45e14T^{2} \)
73 \( 1 + 3.05e7iT - 8.06e14T^{2} \)
79 \( 1 + 6.31e7T + 1.51e15T^{2} \)
83 \( 1 - 3.53e6T + 2.25e15T^{2} \)
89 \( 1 + 9.26e7iT - 3.93e15T^{2} \)
97 \( 1 + 1.20e8iT - 7.83e15T^{2} \)
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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

−11.31241180404559297996817875408, −10.38204692639993734299130428171, −9.198987567649834923845992264428, −8.540383956528154271168775263510, −7.40381321371046444140866014691, −5.68179604257043652864410857861, −4.43035723760893696355752351244, −3.49225026403209812781686498932, −1.80688849567489270858892497376, −0.29364748499525853344023124786, 1.93325578792374327162730280987, 2.56481627935715063391916499425, 4.06459123178604593047208981302, 5.93006662930153589812372790436, 6.97476592782812886535472652875, 7.86995775163613611842491371927, 9.182391350221613175030819844243, 9.974984026261343687568194757438, 11.46205190038354200299643984701, 12.33464919700434860192729093922

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