Properties

Label 2-120-120.59-c1-0-2
Degree $2$
Conductor $120$
Sign $-0.395 - 0.918i$
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  + (0.866 + 1.11i)2-s − 1.73·3-s + (−0.500 + 1.93i)4-s + 2.23i·5-s + (−1.49 − 1.93i)6-s + (−2.59 + 1.11i)8-s + 2.99·9-s + (−2.50 + 1.93i)10-s + (0.866 − 3.35i)12-s − 3.87i·15-s + (−3.5 − 1.93i)16-s + 6.92·17-s + (2.59 + 3.35i)18-s + 4·19-s + (−4.33 − 1.11i)20-s + ⋯
L(s)  = 1  + (0.612 + 0.790i)2-s − 1.00·3-s + (−0.250 + 0.968i)4-s + 0.999i·5-s + (−0.612 − 0.790i)6-s + (−0.918 + 0.395i)8-s + 0.999·9-s + (−0.790 + 0.612i)10-s + (0.250 − 0.968i)12-s − 1.00i·15-s + (−0.875 − 0.484i)16-s + 1.68·17-s + (0.612 + 0.790i)18-s + 0.917·19-s + (−0.968 − 0.250i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.395 - 0.918i$
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: \((2,\ 120,\ (\ :1/2),\ -0.395 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.571002 + 0.867349i\)
\(L(\frac12)\) \(\approx\) \(0.571002 + 0.867349i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 1.11i)T \)
3 \( 1 + 1.73T \)
5 \( 1 - 2.23iT \)
good7 \( 1 + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 6.92T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 8.94iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 7.74iT - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 8.94iT - 47T^{2} \)
53 \( 1 + 4.47iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 15.4iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 7.74iT - 79T^{2} \)
83 \( 1 - 3.46T + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−14.07583105127777397105014190287, −12.67563606505328238744607703081, −11.96579569717191887322583573782, −10.85671098204716585909342961535, −9.772644300205767193251000508362, −7.955335915614753098599944535934, −6.93955839577251695221461704680, −6.05532064796848729882870449626, −4.90396513482860089923114534067, −3.30604674302533284706138136287, 1.24871830823032357325157849446, 3.78898483280470033170864145423, 5.19345208285814782747346326183, 5.79089491580389829527964502812, 7.60429980742977750577878436097, 9.426464548989252731432720387646, 10.09576733678791506577609402592, 11.54758214659964146686361762609, 11.95630402599732234585551874834, 12.97161248345628890331946702714

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