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Properties

Label 2-120-120.59-c1-0-10
Degree $2$
Conductor $120$
Sign $0.965 - 0.259i$
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 + (0.183 + 2.44i)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 + (1.56 + 3.09i)12-s − 4.65·13-s + (4.29 − 1.78i)14-s + (−2.35 − 3.07i)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 + (0.0748 + 0.997i)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 + (0.450 + 0.892i)12-s − 1.29·13-s + (1.14 − 0.475i)14-s + (−0.609 − 0.793i)15-s − i·16-s − 0.896·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.965 - 0.259i$
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: \((2,\ 120,\ (\ :1/2),\ 0.965 - 0.259i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51367 + 0.199512i\)
\(L(\frac12)\) \(\approx\) \(1.51367 + 0.199512i\)
\(L(\frac{3}{2})\) 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

−13.84708498679447777135025673803, −12.13715191625321046908267293130, −11.35062573991602776134712403822, −10.85866047061292968539487998328, −9.736694185519729921740280487918, −8.042207304822723037877938415520, −6.59885521944234893067007730064, −5.17618412761611156655762021222, −4.26023846643999984491931086043, −2.82304005995032521481622582497, 2.12174215673087104247097525244, 4.55646788193322558451110655672, 5.22426025386522193083965894859, 6.91638926960422020293116211718, 7.72961979167501126242858879964, 8.640897529960947201732920776977, 10.83524102693976651293622939478, 12.05662815796850838539957792052, 12.19516989166445017328410954975, 13.37558729152493185464555047848

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