Properties

Label 2-120-120.59-c1-0-5
Degree $2$
Conductor $120$
Sign $0.623 - 0.781i$
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  + (−1.30 + 0.541i)2-s + (0.541 + 1.64i)3-s + (1.41 − 1.41i)4-s + (1.25 − 1.84i)5-s + (−1.59 − 1.85i)6-s + 3.29·7-s + (−1.08 + 2.61i)8-s + (−2.41 + 1.78i)9-s + (−0.645 + 3.09i)10-s + 2.51i·11-s + (3.09 + 1.56i)12-s − 4.65·13-s + (−4.29 + 1.78i)14-s + (3.72 + 1.07i)15-s − 4i·16-s + 3.69·17-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)2-s + (0.312 + 0.949i)3-s + (0.707 − 0.707i)4-s + (0.563 − 0.826i)5-s + (−0.652 − 0.758i)6-s + 1.24·7-s + (−0.382 + 0.923i)8-s + (−0.804 + 0.593i)9-s + (−0.204 + 0.978i)10-s + 0.759i·11-s + (0.892 + 0.450i)12-s − 1.29·13-s + (−1.14 + 0.475i)14-s + (0.960 + 0.276i)15-s i·16-s + 0.896·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.623 - 0.781i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :1/2),\ 0.623 - 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.803040 + 0.386757i\)
\(L(\frac12)\) \(\approx\) \(0.803040 + 0.386757i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 - 0.541i)T \)
3 \( 1 + (-0.541 - 1.64i)T \)
5 \( 1 + (-1.25 + 1.84i)T \)
good7 \( 1 - 3.29T + 7T^{2} \)
11 \( 1 - 2.51iT - 11T^{2} \)
13 \( 1 + 4.65T + 13T^{2} \)
17 \( 1 - 3.69T + 17T^{2} \)
19 \( 1 - 0.828T + 19T^{2} \)
23 \( 1 + 2.61iT - 23T^{2} \)
29 \( 1 + 6.08T + 29T^{2} \)
31 \( 1 + 1.17iT - 31T^{2} \)
37 \( 1 + 1.92T + 37T^{2} \)
41 \( 1 + 8.59iT - 41T^{2} \)
43 \( 1 - 6.01iT - 43T^{2} \)
47 \( 1 - 2.61iT - 47T^{2} \)
53 \( 1 + 4.59iT - 53T^{2} \)
59 \( 1 + 2.51iT - 59T^{2} \)
61 \( 1 + 8.48iT - 61T^{2} \)
67 \( 1 + 3.29iT - 67T^{2} \)
71 \( 1 + 7.12T + 71T^{2} \)
73 \( 1 - 6.58iT - 73T^{2} \)
79 \( 1 - 16.4iT - 79T^{2} \)
83 \( 1 + 9.37T + 83T^{2} \)
89 \( 1 - 5.03iT - 89T^{2} \)
97 \( 1 + 2.72iT - 97T^{2} \)
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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

−14.21766596236399480250282812567, −12.38574940154574820563031536047, −11.26131003303173084221039094904, −10.07332611226343479010296373145, −9.480900968872376155021201484544, −8.402629662360313603637777139628, −7.50829940496155040846872623720, −5.49345642072499723526925121580, −4.73789409013169917097316458723, −2.07608496722704828197574997567, 1.73610263099803189247974954658, 3.06203263807475550417502343270, 5.74269853069801206264339645395, 7.22162352363267510317866073168, 7.80881104697005118075569315287, 9.018234130715638059539196703076, 10.19788936866637195945396117114, 11.33799955974176083622512128209, 11.97870387752774676354175967517, 13.29948416996291836348006270976

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