↑

Properties

Label 2-120-120.77-c1-0-4
Degree $2$
Conductor $120$
Sign $0.922 - 0.386i$
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  + (−1.11 + 0.864i)2-s + (−1.72 + 0.170i)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 + (1.10 + 2.60i)8-s + (2.94 − 0.586i)9-s + (0.00136 + 3.16i)10-s + 0.510·11-s + (−0.543 + 3.42i)12-s + (0.750 + 0.750i)13-s + (−4.10 − 0.528i)14-s + (−2.05 + 3.28i)15-s + (−3.48 − 1.95i)16-s + (3.14 − 3.14i)17-s + ⋯
L(s)  = 1  + (−0.791 + 0.611i)2-s + (−0.995 + 0.0982i)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 + (0.390 + 0.920i)8-s + (0.980 − 0.195i)9-s + (0.000431 + 0.999i)10-s + 0.153·11-s + (−0.156 + 0.987i)12-s + (0.208 + 0.208i)13-s + (−1.09 − 0.141i)14-s + (−0.530 + 0.847i)15-s + (−0.871 − 0.489i)16-s + (0.763 − 0.763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.922 - 0.386i$
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: \((2,\ 120,\ (\ :1/2),\ 0.922 - 0.386i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.663629 + 0.133331i\)
\(L(\frac12)\) \(\approx\) \(0.663629 + 0.133331i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.11 - 0.864i)T \)
3 \( 1 + (1.72 - 0.170i)T \)
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

−13.68786972606031236637635944864, −12.16696743801768725544238459334, −11.50196995411627195043734203116, −10.18612940833491623332282739917, −9.333416937766281183197476701672, −8.262888617966822462811796148459, −6.88279653912566139104850976606, −5.52453110322240153761576597267, −5.06205027851121778206479103521, −1.46181500139400580898830286669, 1.54892640065621460814814545383, 3.75326623023645439672240749277, 5.60976332183223407175488676882, 7.03828300430731751745772822914, 7.84913391667917041842575471714, 9.656642066221351164514791007844, 10.41927163492194875731940166661, 11.16130225214773430872985361331, 11.99067188360925194677272702204, 13.23724069851517596689142995584

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