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Properties

Label 2-120-120.53-c1-0-0
Degree $2$
Conductor $120$
Sign $-0.528 - 0.849i$
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  + (−0.250 − 1.39i)2-s + (−1.40 + 1.01i)3-s + (−1.87 + 0.696i)4-s + (−2.23 − 0.116i)5-s + (1.76 + 1.69i)6-s + (−2.29 + 2.29i)7-s + (1.43 + 2.43i)8-s + (0.925 − 2.85i)9-s + (0.396 + 3.13i)10-s − 2.28·11-s + (1.91 − 2.88i)12-s + (−1.05 + 1.05i)13-s + (3.76 + 2.61i)14-s + (3.24 − 2.11i)15-s + (3.03 − 2.61i)16-s + (−3.04 − 3.04i)17-s + ⋯
L(s)  = 1  + (−0.176 − 0.984i)2-s + (−0.808 + 0.588i)3-s + (−0.937 + 0.348i)4-s + (−0.998 − 0.0519i)5-s + (0.721 + 0.692i)6-s + (−0.865 + 0.865i)7-s + (0.508 + 0.861i)8-s + (0.308 − 0.951i)9-s + (0.125 + 0.992i)10-s − 0.688·11-s + (0.553 − 0.832i)12-s + (−0.292 + 0.292i)13-s + (1.00 + 0.698i)14-s + (0.838 − 0.545i)15-s + (0.757 − 0.652i)16-s + (−0.738 − 0.738i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.528 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.528 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.528 - 0.849i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :1/2),\ -0.528 - 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0684702 + 0.123206i\)
\(L(\frac12)\) \(\approx\) \(0.0684702 + 0.123206i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.250 + 1.39i)T \)
3 \( 1 + (1.40 - 1.01i)T \)
5 \( 1 + (2.23 + 0.116i)T \)
good7 \( 1 + (2.29 - 2.29i)T - 7iT^{2} \)
11 \( 1 + 2.28T + 11T^{2} \)
13 \( 1 + (1.05 - 1.05i)T - 13iT^{2} \)
17 \( 1 + (3.04 + 3.04i)T + 17iT^{2} \)
19 \( 1 - 3.36T + 19T^{2} \)
23 \( 1 + (3.68 - 3.68i)T - 23iT^{2} \)
29 \( 1 + 2.71iT - 29T^{2} \)
31 \( 1 + 6.49T + 31T^{2} \)
37 \( 1 + (2.31 + 2.31i)T + 37iT^{2} \)
41 \( 1 - 10.8iT - 41T^{2} \)
43 \( 1 + (-1.16 + 1.16i)T - 43iT^{2} \)
47 \( 1 + (-1.83 - 1.83i)T + 47iT^{2} \)
53 \( 1 + (-5.82 - 5.82i)T + 53iT^{2} \)
59 \( 1 - 7.41iT - 59T^{2} \)
61 \( 1 - 8.97iT - 61T^{2} \)
67 \( 1 + (8.66 + 8.66i)T + 67iT^{2} \)
71 \( 1 + 7.37iT - 71T^{2} \)
73 \( 1 + (-1.83 - 1.83i)T + 73iT^{2} \)
79 \( 1 + 8.28iT - 79T^{2} \)
83 \( 1 + (5.27 + 5.27i)T + 83iT^{2} \)
89 \( 1 + 11.5T + 89T^{2} \)
97 \( 1 + (2.79 - 2.79i)T - 97iT^{2} \)
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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

−13.42057787139710707378310602216, −12.32159607524222993166377443286, −11.78485816130324788332871914745, −10.90632516119689315065975429600, −9.728587121698653480090017635598, −8.975726000229065391069929330307, −7.42808201063871309776228899893, −5.62738901770170043107414454746, −4.38886028159616276025702354408, −3.05035360483496364605019774316, 0.17326610436264061865409740947, 3.96707316029091078975077733546, 5.36686359855572284247375544399, 6.74555607557517019919787942488, 7.37879489656611302781134256073, 8.413067886657758512271212797499, 10.10836855511168958757901397256, 10.90026504448886863882164235244, 12.43323521982432767576917529711, 13.06722445024430767950643036332

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