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Properties

Label 2-120-15.8-c1-0-0
Degree $2$
Conductor $120$
Sign $-0.374 - 0.927i$
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  + (−1.41 + i)3-s + (−1 + 2i)5-s + (−2.41 + 2.41i)7-s + (1.00 − 2.82i)9-s + 0.828i·11-s + (3.82 + 3.82i)13-s + (−0.585 − 3.82i)15-s + (−1.82 − 1.82i)17-s + 0.828i·19-s + (1 − 5.82i)21-s + (4.41 − 4.41i)23-s + (−3 − 4i)25-s + (1.41 + 5.00i)27-s + 3.65·29-s + 5.65·31-s + ⋯
L(s)  = 1  + (−0.816 + 0.577i)3-s + (−0.447 + 0.894i)5-s + (−0.912 + 0.912i)7-s + (0.333 − 0.942i)9-s + 0.249i·11-s + (1.06 + 1.06i)13-s + (−0.151 − 0.988i)15-s + (−0.443 − 0.443i)17-s + 0.190i·19-s + (0.218 − 1.27i)21-s + (0.920 − 0.920i)23-s + (−0.600 − 0.800i)25-s + (0.272 + 0.962i)27-s + 0.679·29-s + 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.374 - 0.927i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :1/2),\ -0.374 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.370015 + 0.548382i\)
\(L(\frac12)\) \(\approx\) \(0.370015 + 0.548382i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.41 - i)T \)
5 \( 1 + (1 - 2i)T \)
good7 \( 1 + (2.41 - 2.41i)T - 7iT^{2} \)
11 \( 1 - 0.828iT - 11T^{2} \)
13 \( 1 + (-3.82 - 3.82i)T + 13iT^{2} \)
17 \( 1 + (1.82 + 1.82i)T + 17iT^{2} \)
19 \( 1 - 0.828iT - 19T^{2} \)
23 \( 1 + (-4.41 + 4.41i)T - 23iT^{2} \)
29 \( 1 - 3.65T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + (5.82 - 5.82i)T - 37iT^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + (0.414 + 0.414i)T + 43iT^{2} \)
47 \( 1 + (-3.58 - 3.58i)T + 47iT^{2} \)
53 \( 1 + (3 - 3i)T - 53iT^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 0.343T + 61T^{2} \)
67 \( 1 + (-10.0 + 10.0i)T - 67iT^{2} \)
71 \( 1 - 10.4iT - 71T^{2} \)
73 \( 1 + (4.65 + 4.65i)T + 73iT^{2} \)
79 \( 1 + 0.828iT - 79T^{2} \)
83 \( 1 + (-3.24 + 3.24i)T - 83iT^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 + (-1 + i)T - 97iT^{2} \)
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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

−13.85415369877138009610182596733, −12.46567166577837345007024591286, −11.65653227873679593349079923110, −10.80194587694326176831031357925, −9.732839906929769034962417763783, −8.695653315685768804284715080715, −6.74642893262408834765087481994, −6.23665494021807760978154229527, −4.53639411472953260540295672291, −3.08490949477847791697644051963, 0.830984185157060022921602174057, 3.72268575311739549786412265139, 5.25571651160581903451386098816, 6.46407157083440294202153517046, 7.60887851409104569524662869298, 8.748565097446662546464856425687, 10.27927253696713683794689671254, 11.11603432979463330795768851897, 12.29284362261712807725691561983, 13.16297700093102608501233200237

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