↑

Properties

Label 2-120-120.77-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

Related objects

Downloads

Learn more

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: $\chi_{120} (77, \cdot )$
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 \( 1 + (-10.8 - 10.8i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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