Properties

Label 2-120-120.59-c1-0-1
Degree $2$
Conductor $120$
Sign $-0.112 - 0.993i$
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  + (−1.30 − 0.541i)2-s + (0.541 + 1.64i)3-s + (1.41 + 1.41i)4-s + (−1.25 + 1.84i)5-s + (0.183 − 2.44i)6-s − 3.29·7-s + (−1.08 − 2.61i)8-s + (−2.41 + 1.78i)9-s + (2.64 − 1.73i)10-s + 2.51i·11-s + (−1.56 + 3.09i)12-s + 4.65·13-s + (4.29 + 1.78i)14-s + (−3.72 − 1.07i)15-s + 4i·16-s + 3.69·17-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)2-s + (0.312 + 0.949i)3-s + (0.707 + 0.707i)4-s + (−0.563 + 0.826i)5-s + (0.0748 − 0.997i)6-s − 1.24·7-s + (−0.382 − 0.923i)8-s + (−0.804 + 0.593i)9-s + (0.836 − 0.547i)10-s + 0.759i·11-s + (−0.450 + 0.892i)12-s + 1.29·13-s + (1.14 + 0.475i)14-s + (−0.960 − 0.276i)15-s + i·16-s + 0.896·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.112 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.112 - 0.993i$
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: \((2,\ 120,\ (\ :1/2),\ -0.112 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.408894 + 0.457584i\)
\(L(\frac12)\) \(\approx\) \(0.408894 + 0.457584i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 + 0.541i)T \)
3 \( 1 + (-0.541 - 1.64i)T \)
5 \( 1 + (1.25 - 1.84i)T \)
good7 \( 1 + 3.29T + 7T^{2} \)
11 \( 1 - 2.51iT - 11T^{2} \)
13 \( 1 - 4.65T + 13T^{2} \)
17 \( 1 - 3.69T + 17T^{2} \)
19 \( 1 - 0.828T + 19T^{2} \)
23 \( 1 - 2.61iT - 23T^{2} \)
29 \( 1 - 6.08T + 29T^{2} \)
31 \( 1 - 1.17iT - 31T^{2} \)
37 \( 1 - 1.92T + 37T^{2} \)
41 \( 1 + 8.59iT - 41T^{2} \)
43 \( 1 - 6.01iT - 43T^{2} \)
47 \( 1 + 2.61iT - 47T^{2} \)
53 \( 1 - 4.59iT - 53T^{2} \)
59 \( 1 + 2.51iT - 59T^{2} \)
61 \( 1 - 8.48iT - 61T^{2} \)
67 \( 1 + 3.29iT - 67T^{2} \)
71 \( 1 - 7.12T + 71T^{2} \)
73 \( 1 - 6.58iT - 73T^{2} \)
79 \( 1 + 16.4iT - 79T^{2} \)
83 \( 1 + 9.37T + 83T^{2} \)
89 \( 1 - 5.03iT - 89T^{2} \)
97 \( 1 + 2.72iT - 97T^{2} \)
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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

−13.80434361386773242921087973744, −12.43375405022231617266504554302, −11.35618522044062298508909402618, −10.36534437260331951469417653580, −9.775917967362316787103410627326, −8.655969790533462698900268322225, −7.46318497893329638135211601259, −6.21997919624154830345733367663, −3.82837918064377410292606390095, −2.97194048793232155159349602487, 0.900382247523387602371549213780, 3.24691575059164648916771440846, 5.81069524151624465652302647658, 6.68867888071504281896100428092, 8.035279295185481705588386758815, 8.648746066818460185825467679494, 9.698826031970218220265434518231, 11.17528248020064573366001684043, 12.20116637610163682661251364066, 13.15308067929883444907106558607

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