Properties

Label 2-120-120.59-c1-0-13
Degree $2$
Conductor $120$
Sign $0.782 + 0.623i$
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 + (2.13 − 2.33i)10-s − 4.20i·11-s + (−0.239 − 3.45i)12-s − 3.21·13-s + (−1.23 + 2.97i)14-s + (1.87 + 3.38i)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 + (0.675 − 0.737i)10-s − 1.26i·11-s + (−0.0692 − 0.997i)12-s − 0.891·13-s + (−0.328 + 0.794i)14-s + (0.484 + 0.875i)15-s + i·16-s + 0.371·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 + 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.782 + 0.623i$
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: \((2,\ 120,\ (\ :1/2),\ 0.782 + 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38754 - 0.485202i\)
\(L(\frac12)\) \(\approx\) \(1.38754 - 0.485202i\)
\(L(\frac{3}{2})\) 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

−13.42517704310427999401719126874, −12.58790517604808832053006694507, −11.01157329627151345804455378008, −10.20744571437767667737739833959, −9.480476106158081012508620652035, −8.495596996186060220843132942190, −6.39932684106204987704310751492, −5.12056728808378425870375499291, −3.50207224093443734710015805363, −2.48009833454209043096434564647, 2.55484236272690200111100406747, 4.41457162137658854128575715410, 6.01962328633680309308725454851, 6.92700848749489147962584129560, 7.986851897594678504804044574208, 9.356721776099003174038156149692, 9.791236946817159502717927280096, 12.32392081998281347490556967045, 12.79975771513327906528693383002, 13.56316757210808423489090171784

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