Properties

Label 2-120-40.29-c1-0-10
Degree $2$
Conductor $120$
Sign $0.742 + 0.669i$
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  + (1.16 − 0.806i)2-s + 3-s + (0.699 − 1.87i)4-s + (−1.86 + 1.23i)5-s + (1.16 − 0.806i)6-s − 0.746i·7-s + (−0.699 − 2.74i)8-s + 9-s + (−1.16 + 2.94i)10-s + 5.36i·11-s + (0.699 − 1.87i)12-s − 2.92·13-s + (−0.601 − 0.866i)14-s + (−1.86 + 1.23i)15-s + (−3.02 − 2.62i)16-s − 2.13i·17-s + ⋯
L(s)  = 1  + (0.821 − 0.570i)2-s + 0.577·3-s + (0.349 − 0.936i)4-s + (−0.832 + 0.554i)5-s + (0.474 − 0.329i)6-s − 0.282i·7-s + (−0.247 − 0.968i)8-s + 0.333·9-s + (−0.367 + 0.930i)10-s + 1.61i·11-s + (0.201 − 0.540i)12-s − 0.811·13-s + (−0.160 − 0.231i)14-s + (−0.480 + 0.320i)15-s + (−0.755 − 0.655i)16-s − 0.517i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.742 + 0.669i$
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: \((2,\ 120,\ (\ :1/2),\ 0.742 + 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52427 - 0.585313i\)
\(L(\frac12)\) \(\approx\) \(1.52427 - 0.585313i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.16 + 0.806i)T \)
3 \( 1 - T \)
5 \( 1 + (1.86 - 1.23i)T \)
good7 \( 1 + 0.746iT - 7T^{2} \)
11 \( 1 - 5.36iT - 11T^{2} \)
13 \( 1 + 2.92T + 13T^{2} \)
17 \( 1 + 2.13iT - 17T^{2} \)
19 \( 1 + 1.73iT - 19T^{2} \)
23 \( 1 - 7.49iT - 23T^{2} \)
29 \( 1 + 6.74iT - 29T^{2} \)
31 \( 1 - 2.64T + 31T^{2} \)
37 \( 1 + 1.07T + 37T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 - 7.44T + 43T^{2} \)
47 \( 1 + 1.73iT - 47T^{2} \)
53 \( 1 - 7.72T + 53T^{2} \)
59 \( 1 + 6.85iT - 59T^{2} \)
61 \( 1 + 6.45iT - 61T^{2} \)
67 \( 1 + 7.44T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + 0.690iT - 73T^{2} \)
79 \( 1 + 2.64T + 79T^{2} \)
83 \( 1 + 5.85T + 83T^{2} \)
89 \( 1 + 7.59T + 89T^{2} \)
97 \( 1 - 14.1iT - 97T^{2} \)
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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

−13.40707307928585619874546719501, −12.25235867703282285439072301782, −11.59087369458695607178585549719, −10.26493136790318114127519359148, −9.516420287208682328702926803208, −7.60251564606655788141626293867, −6.89671113950949232112588771456, −4.90721198771548058073980933131, −3.82509730690766540785535225785, −2.38843319424567991974975107363, 3.01815391503721367543156529001, 4.25814660963111476603875536677, 5.58983139425301669227575689789, 7.02996087629922503283251163160, 8.313065227256583108289902482042, 8.724905221147239609769455408057, 10.71591785328067054785869939277, 11.96189783251180949984834350717, 12.64175494394524121762673690950, 13.71922693726865187756865063086

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