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Properties

Label 2-120-120.77-c1-0-13
Degree $2$
Conductor $120$
Sign $0.980 - 0.195i$
Analytic cond. $0.958204$
Root an. cond. $0.978879$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.39 − 0.250i)2-s + (1.01 + 1.40i)3-s + (1.87 − 0.696i)4-s + (−2.23 + 0.116i)5-s + (1.76 + 1.69i)6-s + (−2.29 − 2.29i)7-s + (2.43 − 1.43i)8-s + (−0.925 + 2.85i)9-s + (−3.07 + 0.720i)10-s − 2.28·11-s + (2.88 + 1.91i)12-s + (1.05 + 1.05i)13-s + (−3.76 − 2.61i)14-s + (−2.43 − 3.00i)15-s + (3.03 − 2.61i)16-s + (3.04 − 3.04i)17-s + ⋯
L(s)  = 1  + (0.984 − 0.176i)2-s + (0.588 + 0.808i)3-s + (0.937 − 0.348i)4-s + (−0.998 + 0.0519i)5-s + (0.721 + 0.692i)6-s + (−0.865 − 0.865i)7-s + (0.861 − 0.508i)8-s + (−0.308 + 0.951i)9-s + (−0.973 + 0.227i)10-s − 0.688·11-s + (0.832 + 0.553i)12-s + (0.292 + 0.292i)13-s + (−1.00 − 0.698i)14-s + (−0.629 − 0.777i)15-s + (0.757 − 0.652i)16-s + (0.738 − 0.738i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.980 - 0.195i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :1/2),\ 0.980 - 0.195i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73329 + 0.170982i\)
\(L(\frac12)\) \(\approx\) \(1.73329 + 0.170982i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.250i)T \)
3 \( 1 + (-1.01 - 1.40i)T \)
5 \( 1 + (2.23 - 0.116i)T \)
good7 \( 1 + (2.29 + 2.29i)T + 7iT^{2} \)
11 \( 1 + 2.28T + 11T^{2} \)
13 \( 1 + (-1.05 - 1.05i)T + 13iT^{2} \)
17 \( 1 + (-3.04 + 3.04i)T - 17iT^{2} \)
19 \( 1 + 3.36T + 19T^{2} \)
23 \( 1 + (-3.68 - 3.68i)T + 23iT^{2} \)
29 \( 1 - 2.71iT - 29T^{2} \)
31 \( 1 + 6.49T + 31T^{2} \)
37 \( 1 + (-2.31 + 2.31i)T - 37iT^{2} \)
41 \( 1 - 10.8iT - 41T^{2} \)
43 \( 1 + (1.16 + 1.16i)T + 43iT^{2} \)
47 \( 1 + (1.83 - 1.83i)T - 47iT^{2} \)
53 \( 1 + (-5.82 + 5.82i)T - 53iT^{2} \)
59 \( 1 + 7.41iT - 59T^{2} \)
61 \( 1 - 8.97iT - 61T^{2} \)
67 \( 1 + (-8.66 + 8.66i)T - 67iT^{2} \)
71 \( 1 + 7.37iT - 71T^{2} \)
73 \( 1 + (-1.83 + 1.83i)T - 73iT^{2} \)
79 \( 1 - 8.28iT - 79T^{2} \)
83 \( 1 + (5.27 - 5.27i)T - 83iT^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 + (2.79 + 2.79i)T + 97iT^{2} \)
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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

−13.49952811692865906718761726837, −12.75230091495172076399976129348, −11.33953921043298250974351354973, −10.62398401980120016245846460735, −9.544880709057562103681074038640, −7.910558127086609728157796719407, −6.92617174673655988098345879685, −5.13278063036350406261352142616, −3.92096075349122477689781910497, −3.09713110015318820756830541090, 2.64982127043694289258862022966, 3.77338372295083905536486703812, 5.64412936629592742128139491280, 6.77583212803665820241976890697, 7.87092181815821649738201924799, 8.768900006681544833093619218259, 10.63643924654515772583486820642, 11.92567997588791401248898200788, 12.68611457031687562516319739614, 13.10833475658546002814765516394

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