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Properties

Label 2-120-120.59-c1-0-7
Degree $2$
Conductor $120$
Sign $0.999 - 0.0258i$
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  + (−0.541 + 1.30i)2-s + (−1.30 − 1.13i)3-s + (−1.41 − 1.41i)4-s + (2.10 − 0.765i)5-s + (2.19 − 1.09i)6-s + 2.27·7-s + (2.61 − 1.08i)8-s + (0.414 + 2.97i)9-s + (−0.137 + 3.15i)10-s − 4.20i·11-s + (0.239 + 3.45i)12-s + 3.21·13-s + (−1.23 + 2.97i)14-s + (−3.61 − 1.38i)15-s + 4i·16-s − 1.53·17-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)2-s + (−0.754 − 0.656i)3-s + (−0.707 − 0.707i)4-s + (0.939 − 0.342i)5-s + (0.895 − 0.445i)6-s + 0.859·7-s + (0.923 − 0.382i)8-s + (0.138 + 0.990i)9-s + (−0.0433 + 0.999i)10-s − 1.26i·11-s + (0.0692 + 0.997i)12-s + 0.891·13-s + (−0.328 + 0.794i)14-s + (−0.933 − 0.358i)15-s + i·16-s − 0.371·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.999 - 0.0258i$
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: \((2,\ 120,\ (\ :1/2),\ 0.999 - 0.0258i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.820313 + 0.0106248i\)
\(L(\frac12)\) \(\approx\) \(0.820313 + 0.0106248i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.541 - 1.30i)T \)
3 \( 1 + (1.30 + 1.13i)T \)
5 \( 1 + (-2.10 + 0.765i)T \)
good7 \( 1 - 2.27T + 7T^{2} \)
11 \( 1 + 4.20iT - 11T^{2} \)
13 \( 1 - 3.21T + 13T^{2} \)
17 \( 1 + 1.53T + 17T^{2} \)
19 \( 1 + 4.82T + 19T^{2} \)
23 \( 1 - 1.08iT - 23T^{2} \)
29 \( 1 - 1.74T + 29T^{2} \)
31 \( 1 - 6.82iT - 31T^{2} \)
37 \( 1 + 7.76T + 37T^{2} \)
41 \( 1 - 2.46iT - 41T^{2} \)
43 \( 1 - 8.70iT - 43T^{2} \)
47 \( 1 + 1.08iT - 47T^{2} \)
53 \( 1 - 11.0iT - 53T^{2} \)
59 \( 1 - 4.20iT - 59T^{2} \)
61 \( 1 + 8.48iT - 61T^{2} \)
67 \( 1 - 2.27iT - 67T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 + 4.54iT - 73T^{2} \)
79 \( 1 - 0.485iT - 79T^{2} \)
83 \( 1 + 6.94T + 83T^{2} \)
89 \( 1 + 8.40iT - 89T^{2} \)
97 \( 1 + 10.9iT - 97T^{2} \)
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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

−13.67748840865053941871368728651, −12.75638490714412374212304603952, −11.16197325047121520967916377058, −10.47389116879753123096417621485, −8.827031563412472502174455876305, −8.167377836369788579135676450107, −6.61144246397733502311737317836, −5.86483301340940991109522612530, −4.82372382416558566630090639772, −1.40528954866311429447308415711, 1.96727232681080610180130496397, 4.08253443869356656093441159910, 5.22628428412587413209462470704, 6.78267348044292140953102581127, 8.529725951152028105236274280592, 9.612723851433688141189818207374, 10.48646937256907386255027701191, 11.12332821455227990014593122777, 12.22024823206813452893799223038, 13.22176851571218176670011607136

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