Properties

Label 2-120-120.53-c1-0-11
Degree $2$
Conductor $120$
Sign $0.823 - 0.567i$
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  + (0.864 + 1.11i)2-s + (−0.170 − 1.72i)3-s + (−0.506 + 1.93i)4-s + (1.36 + 1.77i)5-s + (1.78 − 1.68i)6-s + (2.06 − 2.06i)7-s + (−2.60 + 1.10i)8-s + (−2.94 + 0.586i)9-s + (−0.801 + 3.05i)10-s + 0.510·11-s + (3.42 + 0.543i)12-s + (−0.750 + 0.750i)13-s + (4.10 + 0.528i)14-s + (2.81 − 2.65i)15-s + (−3.48 − 1.95i)16-s + (−3.14 − 3.14i)17-s + ⋯
L(s)  = 1  + (0.611 + 0.791i)2-s + (−0.0982 − 0.995i)3-s + (−0.253 + 0.967i)4-s + (0.610 + 0.791i)5-s + (0.727 − 0.685i)6-s + (0.782 − 0.782i)7-s + (−0.920 + 0.390i)8-s + (−0.980 + 0.195i)9-s + (−0.253 + 0.967i)10-s + 0.153·11-s + (0.987 + 0.156i)12-s + (−0.208 + 0.208i)13-s + (1.09 + 0.141i)14-s + (0.727 − 0.685i)15-s + (−0.871 − 0.489i)16-s + (−0.763 − 0.763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.823 - 0.567i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :1/2),\ 0.823 - 0.567i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35722 + 0.422439i\)
\(L(\frac12)\) \(\approx\) \(1.35722 + 0.422439i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.864 - 1.11i)T \)
3 \( 1 + (0.170 + 1.72i)T \)
5 \( 1 + (-1.36 - 1.77i)T \)
good7 \( 1 + (-2.06 + 2.06i)T - 7iT^{2} \)
11 \( 1 - 0.510T + 11T^{2} \)
13 \( 1 + (0.750 - 0.750i)T - 13iT^{2} \)
17 \( 1 + (3.14 + 3.14i)T + 17iT^{2} \)
19 \( 1 + 6.01T + 19T^{2} \)
23 \( 1 + (-2.54 + 2.54i)T - 23iT^{2} \)
29 \( 1 + 5.10iT - 29T^{2} \)
31 \( 1 + 4.56T + 31T^{2} \)
37 \( 1 + (-6.76 - 6.76i)T + 37iT^{2} \)
41 \( 1 - 4.24iT - 41T^{2} \)
43 \( 1 + (-5.95 + 5.95i)T - 43iT^{2} \)
47 \( 1 + (-3.33 - 3.33i)T + 47iT^{2} \)
53 \( 1 + (-5.75 - 5.75i)T + 53iT^{2} \)
59 \( 1 + 1.16iT - 59T^{2} \)
61 \( 1 - 4.92iT - 61T^{2} \)
67 \( 1 + (7.98 + 7.98i)T + 67iT^{2} \)
71 \( 1 - 5.09iT - 71T^{2} \)
73 \( 1 + (3.20 + 3.20i)T + 73iT^{2} \)
79 \( 1 - 7.31iT - 79T^{2} \)
83 \( 1 + (4.77 + 4.77i)T + 83iT^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 + (-10.8 + 10.8i)T - 97iT^{2} \)
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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

−13.67540791930482703181496074789, −12.97080843975498533608529359840, −11.65651610122125472525716735371, −10.78811045589263048830284725290, −8.976389272167292649395912116850, −7.71240710796889488798012863942, −6.90918133160614802330539274415, −6.02927741538781841300071470527, −4.49210943936387844321570774891, −2.48784412926148606213453331789, 2.18896722262646167671436377768, 4.13030824316579495981267639023, 5.14235069280308496944477148567, 5.99777749288117930972477816758, 8.670796843101778038970087456904, 9.197244997707496805877438094663, 10.47808403693170251966087801640, 11.23527110658258794182789427076, 12.36902443160339588285825973429, 13.19317490945653454110836253682

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