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Properties

Label 2-120-40.27-c1-0-7
Degree $2$
Conductor $120$
Sign $0.921 + 0.389i$
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  + (−0.909 + 1.08i)2-s + (0.707 − 0.707i)3-s + (−0.345 − 1.96i)4-s + (−0.780 − 2.09i)5-s + (0.122 + 1.40i)6-s + (2.10 − 2.10i)7-s + (2.44 + 1.41i)8-s − 1.00i·9-s + (2.97 + 1.05i)10-s − 3.11·11-s + (−1.63 − 1.14i)12-s + (2.19 + 2.19i)13-s + (0.366 + 4.20i)14-s + (−2.03 − 0.929i)15-s + (−3.76 + 1.36i)16-s + (5.48 + 5.48i)17-s + ⋯
L(s)  = 1  + (−0.643 + 0.765i)2-s + (0.408 − 0.408i)3-s + (−0.172 − 0.984i)4-s + (−0.349 − 0.937i)5-s + (0.0501 + 0.575i)6-s + (0.796 − 0.796i)7-s + (0.865 + 0.500i)8-s − 0.333i·9-s + (0.942 + 0.335i)10-s − 0.940·11-s + (−0.472 − 0.331i)12-s + (0.608 + 0.608i)13-s + (0.0978 + 1.12i)14-s + (−0.525 − 0.240i)15-s + (−0.940 + 0.340i)16-s + (1.33 + 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.921 + 0.389i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :1/2),\ 0.921 + 0.389i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.847474 - 0.171683i\)
\(L(\frac12)\) \(\approx\) \(0.847474 - 0.171683i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.909 - 1.08i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.780 + 2.09i)T \)
good7 \( 1 + (-2.10 + 2.10i)T - 7iT^{2} \)
11 \( 1 + 3.11T + 11T^{2} \)
13 \( 1 + (-2.19 - 2.19i)T + 13iT^{2} \)
17 \( 1 + (-5.48 - 5.48i)T + 17iT^{2} \)
19 \( 1 + 3.91iT - 19T^{2} \)
23 \( 1 + (2.42 + 2.42i)T + 23iT^{2} \)
29 \( 1 + 2.23T + 29T^{2} \)
31 \( 1 - 7.17iT - 31T^{2} \)
37 \( 1 + (1.23 - 1.23i)T - 37iT^{2} \)
41 \( 1 - 4.40T + 41T^{2} \)
43 \( 1 + (1.50 - 1.50i)T - 43iT^{2} \)
47 \( 1 + (-2.00 + 2.00i)T - 47iT^{2} \)
53 \( 1 + (-5.56 - 5.56i)T + 53iT^{2} \)
59 \( 1 + 5.44iT - 59T^{2} \)
61 \( 1 - 7.46iT - 61T^{2} \)
67 \( 1 + (-6.40 - 6.40i)T + 67iT^{2} \)
71 \( 1 - 2.00iT - 71T^{2} \)
73 \( 1 + (1.49 - 1.49i)T - 73iT^{2} \)
79 \( 1 + 1.91T + 79T^{2} \)
83 \( 1 + (6.67 - 6.67i)T - 83iT^{2} \)
89 \( 1 + 10.0iT - 89T^{2} \)
97 \( 1 + (10.0 + 10.0i)T + 97iT^{2} \)
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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

−13.62810350947547434718473303891, −12.59372042513882115890613951989, −11.14621559298546969589195558398, −10.11347640663219001731426971404, −8.691939050470656987716984550820, −8.110234171721577963267387552294, −7.19259738608896628094744542188, −5.60603005417975769681907706093, −4.29142868858834982646578167778, −1.35226406366422577507800246869, 2.45710973859606072097448790216, 3.62706926781786311698783040550, 5.45489602084253897042396202450, 7.66031002455143803542407401730, 8.093844182815854316884248131063, 9.532055515888757232742252986709, 10.42023615557365866031826072226, 11.34073744708912367330183949655, 12.15101417620809298613712237404, 13.53189328056162940014480804596

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