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Properties

Label 2-120-120.59-c3-0-46
Degree $2$
Conductor $120$
Sign $0.899 + 0.437i$
Analytic cond. $7.08022$
Root an. cond. $2.66087$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.22 + 1.74i)2-s + (4.99 − 1.43i)3-s + (1.93 − 7.76i)4-s + (10.2 − 4.55i)5-s + (−8.63 + 11.8i)6-s − 3.90·7-s + (9.22 + 20.6i)8-s + (22.9 − 14.2i)9-s + (−14.8 + 27.9i)10-s − 59.1i·11-s + (−1.45 − 41.5i)12-s − 63.6·13-s + (8.70 − 6.80i)14-s + (44.5 − 37.3i)15-s + (−56.5 − 29.9i)16-s + 69.6·17-s + ⋯
L(s)  = 1  + (−0.787 + 0.615i)2-s + (0.961 − 0.275i)3-s + (0.241 − 0.970i)4-s + (0.913 − 0.407i)5-s + (−0.587 + 0.808i)6-s − 0.210·7-s + (0.407 + 0.913i)8-s + (0.848 − 0.529i)9-s + (−0.468 + 0.883i)10-s − 1.62i·11-s + (−0.0350 − 0.999i)12-s − 1.35·13-s + (0.166 − 0.129i)14-s + (0.765 − 0.642i)15-s + (−0.883 − 0.468i)16-s + 0.993·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.899 + 0.437i$
Analytic conductor: \(7.08022\)
Root analytic conductor: \(2.66087\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :3/2),\ 0.899 + 0.437i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.63700 - 0.377150i\)
\(L(\frac12)\) \(\approx\) \(1.63700 - 0.377150i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.22 - 1.74i)T \)
3 \( 1 + (-4.99 + 1.43i)T \)
5 \( 1 + (-10.2 + 4.55i)T \)
good7 \( 1 + 3.90T + 343T^{2} \)
11 \( 1 + 59.1iT - 1.33e3T^{2} \)
13 \( 1 + 63.6T + 2.19e3T^{2} \)
17 \( 1 - 69.6T + 4.91e3T^{2} \)
19 \( 1 - 33.0T + 6.85e3T^{2} \)
23 \( 1 - 90.6iT - 1.21e4T^{2} \)
29 \( 1 - 172.T + 2.43e4T^{2} \)
31 \( 1 - 61.7iT - 2.97e4T^{2} \)
37 \( 1 + 10.7T + 5.06e4T^{2} \)
41 \( 1 - 475. iT - 6.89e4T^{2} \)
43 \( 1 + 59.3iT - 7.95e4T^{2} \)
47 \( 1 + 500. iT - 1.03e5T^{2} \)
53 \( 1 - 407. iT - 1.48e5T^{2} \)
59 \( 1 + 17.5iT - 2.05e5T^{2} \)
61 \( 1 - 245. iT - 2.26e5T^{2} \)
67 \( 1 + 35.7iT - 3.00e5T^{2} \)
71 \( 1 + 889.T + 3.57e5T^{2} \)
73 \( 1 - 617. iT - 3.89e5T^{2} \)
79 \( 1 - 108. iT - 4.93e5T^{2} \)
83 \( 1 - 628.T + 5.71e5T^{2} \)
89 \( 1 - 763. iT - 7.04e5T^{2} \)
97 \( 1 - 866. iT - 9.12e5T^{2} \)
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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

−13.33681925814338100377030419994, −11.92919976279237434685675203728, −10.25096383244273982131327264885, −9.561626541520517746616881881407, −8.632919703550976925991612707331, −7.69891618505883990576151267931, −6.42120987938071030070196638044, −5.23016225755894433842262366308, −2.84992829403550152452827838066, −1.14209744345640055116672406334, 1.93594762501469443151782267647, 2.93539915300381312828840112577, 4.67435938180775282412564267015, 6.93630519622823263543105364278, 7.75192951867439789143875339914, 9.225869662039419968322632749915, 9.904106841577495052725996698224, 10.38133382949822126844191902628, 12.19268792440616040104525666688, 12.85845953116340891770256258454

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