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Properties

Label 2-120-40.29-c1-0-9
Degree $2$
Conductor $120$
Sign $0.756 + 0.654i$
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  + (1.34 − 0.450i)2-s − 3-s + (1.59 − 1.20i)4-s + (0.254 − 2.22i)5-s + (−1.34 + 0.450i)6-s + 2.64i·7-s + (1.59 − 2.33i)8-s + 9-s + (−0.659 − 3.09i)10-s + 1.51i·11-s + (−1.59 + 1.20i)12-s − 3.87·13-s + (1.18 + 3.54i)14-s + (−0.254 + 2.22i)15-s + (1.08 − 3.84i)16-s + 3.31i·17-s + ⋯
L(s)  = 1  + (0.947 − 0.318i)2-s − 0.577·3-s + (0.797 − 0.603i)4-s + (0.113 − 0.993i)5-s + (−0.547 + 0.183i)6-s + 0.998i·7-s + (0.563 − 0.825i)8-s + 0.333·9-s + (−0.208 − 0.978i)10-s + 0.456i·11-s + (−0.460 + 0.348i)12-s − 1.07·13-s + (0.317 + 0.946i)14-s + (−0.0656 + 0.573i)15-s + (0.271 − 0.962i)16-s + 0.803i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.756 + 0.654i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :1/2),\ 0.756 + 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40525 - 0.523259i\)
\(L(\frac12)\) \(\approx\) \(1.40525 - 0.523259i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 + 0.450i)T \)
3 \( 1 + T \)
5 \( 1 + (-0.254 + 2.22i)T \)
good7 \( 1 - 2.64iT - 7T^{2} \)
11 \( 1 - 1.51iT - 11T^{2} \)
13 \( 1 + 3.87T + 13T^{2} \)
17 \( 1 - 3.31iT - 17T^{2} \)
19 \( 1 - 7.08iT - 19T^{2} \)
23 \( 1 + 4.82iT - 23T^{2} \)
29 \( 1 + 2.18iT - 29T^{2} \)
31 \( 1 + 7.36T + 31T^{2} \)
37 \( 1 - 7.87T + 37T^{2} \)
41 \( 1 - 8.72T + 41T^{2} \)
43 \( 1 + 1.01T + 43T^{2} \)
47 \( 1 + 7.08iT - 47T^{2} \)
53 \( 1 + 4.50T + 53T^{2} \)
59 \( 1 + 6.79iT - 59T^{2} \)
61 \( 1 - 3.60iT - 61T^{2} \)
67 \( 1 - 1.01T + 67T^{2} \)
71 \( 1 + 6.72T + 71T^{2} \)
73 \( 1 + 15.5iT - 73T^{2} \)
79 \( 1 - 7.36T + 79T^{2} \)
83 \( 1 + 7.74T + 83T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 - 11.1iT - 97T^{2} \)
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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

−12.85215749268182238244952741735, −12.47746165405455465672646929024, −11.76372014877420549778640051255, −10.38676535116238017890945237812, −9.387682569787316647057491080489, −7.81777905856824308553279539793, −6.15148969379611355540884927692, −5.33880860225500696184261446878, −4.22098229968495723923824090517, −2.04128664062871564010657097566, 2.87615994388096863389644470442, 4.39508971616457155186481528911, 5.69149156033938695983616953089, 7.05349848888652327567376920367, 7.41723458285377844873052520256, 9.637646177083581400572635359735, 11.02618866982096631940481939745, 11.32529500911265445738641507066, 12.76264528448803874740282446109, 13.70209765636108661839520119753

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