↑

Properties

Label 2-120-120.53-c3-0-40
Degree $2$
Conductor $120$
Sign $0.722 + 0.691i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.22 + 1.74i)2-s + (1.34 − 5.01i)3-s + (1.88 − 7.77i)4-s + (10.3 + 4.10i)5-s + (5.78 + 13.5i)6-s + (2.75 − 2.75i)7-s + (9.39 + 20.5i)8-s + (−23.3 − 13.5i)9-s + (−30.2 + 9.05i)10-s + 18.4·11-s + (−36.4 − 19.9i)12-s + (22.4 − 22.4i)13-s + (−1.31 + 10.9i)14-s + (34.5 − 46.6i)15-s + (−56.8 − 29.3i)16-s + (−73.3 − 73.3i)17-s + ⋯
L(s)  = 1  + (−0.786 + 0.618i)2-s + (0.258 − 0.965i)3-s + (0.236 − 0.971i)4-s + (0.930 + 0.367i)5-s + (0.393 + 0.919i)6-s + (0.148 − 0.148i)7-s + (0.415 + 0.909i)8-s + (−0.865 − 0.500i)9-s + (−0.958 + 0.286i)10-s + 0.504·11-s + (−0.877 − 0.479i)12-s + (0.478 − 0.478i)13-s + (−0.0250 + 0.208i)14-s + (0.595 − 0.803i)15-s + (−0.888 − 0.458i)16-s + (−1.04 − 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.722 + 0.691i$
Analytic conductor: \(7.08022\)
Root analytic conductor: \(2.66087\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :3/2),\ 0.722 + 0.691i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.28613 - 0.516296i\)
\(L(\frac12)\) \(\approx\) \(1.28613 - 0.516296i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.34 + 5.01i)T \)
5 \( 1 + (-10.3 - 4.10i)T \)
good7 \( 1 + (-2.75 + 2.75i)T - 343iT^{2} \)
11 \( 1 - 18.4T + 1.33e3T^{2} \)
13 \( 1 + (-22.4 + 22.4i)T - 2.19e3iT^{2} \)
17 \( 1 + (73.3 + 73.3i)T + 4.91e3iT^{2} \)
19 \( 1 - 86.5T + 6.85e3T^{2} \)
23 \( 1 + (-102. + 102. i)T - 1.21e4iT^{2} \)
29 \( 1 + 28.7iT - 2.43e4T^{2} \)
31 \( 1 - 140.T + 2.97e4T^{2} \)
37 \( 1 + (242. + 242. i)T + 5.06e4iT^{2} \)
41 \( 1 + 52.3iT - 6.89e4T^{2} \)
43 \( 1 + (16.8 - 16.8i)T - 7.95e4iT^{2} \)
47 \( 1 + (298. + 298. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-49.3 - 49.3i)T + 1.48e5iT^{2} \)
59 \( 1 - 361. iT - 2.05e5T^{2} \)
61 \( 1 - 437. iT - 2.26e5T^{2} \)
67 \( 1 + (-754. - 754. i)T + 3.00e5iT^{2} \)
71 \( 1 - 554. iT - 3.57e5T^{2} \)
73 \( 1 + (564. + 564. i)T + 3.89e5iT^{2} \)
79 \( 1 - 621. iT - 4.93e5T^{2} \)
83 \( 1 + (-569. - 569. i)T + 5.71e5iT^{2} \)
89 \( 1 - 412.T + 7.04e5T^{2} \)
97 \( 1 + (653. - 653. i)T - 9.12e5iT^{2} \)
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

−13.24787852608496276434837831736, −11.67822585670299031660426896155, −10.65810098434961579169637984615, −9.383999513095686745088965669021, −8.596740202453354593699811365274, −7.22619107384198812247499798490, −6.55537630254491392519749826086, −5.36586092517497772779553086927, −2.56022459497695182852188801658, −1.02228063156381216524218753183, 1.70227309349962013048501270453, 3.36841834654090656333125709920, 4.84480404106722713642841456167, 6.48928236069212048964770682377, 8.316126963835814612901255019645, 9.110368249834811458660029474468, 9.815360187264139890074832235742, 10.86391137435729807796113849974, 11.73378687776385411516384690854, 13.14323779089113085124926557646

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