Properties

Label 2-120-120.59-c1-0-16
Degree $2$
Conductor $120$
Sign $0.499 + 0.866i$
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 + (2.64 + 1.73i)10-s + 2.51i·11-s + (−3.09 − 1.56i)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 + (−0.652 − 0.758i)6-s − 1.24·7-s + (0.382 − 0.923i)8-s + (−0.804 + 0.593i)9-s + (0.836 + 0.547i)10-s + 0.759i·11-s + (−0.892 − 0.450i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.499 + 0.866i$
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: \((2,\ 120,\ (\ :1/2),\ 0.499 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33994 - 0.773886i\)
\(L(\frac12)\) \(\approx\) \(1.33994 - 0.773886i\)
\(L(\frac{3}{2})\) 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

−13.33761635717045763598720842897, −12.55052807930329813822125317494, −11.36937843464211649078378111476, −10.55051007138240382507141372839, −9.339017263736030275639144478923, −7.23209975231332345957134162247, −6.48188703773792509111590339537, −5.65830789690905855248975594480, −3.54179762518627065481637119395, −2.10943270936933276531839013596, 3.22249956800353534867804659320, 4.42741160417538150296028343842, 5.81387765548919653004957860550, 6.37104874028319363874590334363, 8.476278288091162100926259460114, 9.364532668483405125757916021815, 10.69188950686124340811532934847, 11.67061278243001089490208566455, 13.01764294211238761946548685894, 13.41394666130375005076358764493

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