↑

Properties

Label 2-720-5.4-c3-0-31
Degree $2$
Conductor $720$
Sign $i$
Analytic cond. $42.4813$
Root an. cond. $6.51777$
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  + 11.1i·5-s − 138. i·17-s − 164·19-s + 98.3i·23-s − 125.·25-s + 232·31-s − 545. i·47-s + 343·49-s − 621. i·53-s − 358·61-s + 304·79-s − 1.27e3i·83-s + 1.55e3·85-s − 1.83e3i·95-s + 17.8i·107-s + ⋯
L(s)  = 1  + 0.999i·5-s − 1.97i·17-s − 1.98·19-s + 0.891i·23-s − 1.00·25-s + 1.34·31-s − 1.69i·47-s + 49-s − 1.61i·53-s − 0.751·61-s + 0.432·79-s − 1.67i·83-s + 1.97·85-s − 1.98i·95-s + 0.0161i·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $i$
Analytic conductor: \(42.4813\)
Root analytic conductor: \(6.51777\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :3/2),\ i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9662318029\)
\(L(\frac12)\) \(\approx\) \(0.9662318029\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - 11.1iT \)
good7 \( 1 - 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 + 138. iT - 4.91e3T^{2} \)
19 \( 1 + 164T + 6.85e3T^{2} \)
23 \( 1 - 98.3iT - 1.21e4T^{2} \)
29 \( 1 + 2.43e4T^{2} \)
31 \( 1 - 232T + 2.97e4T^{2} \)
37 \( 1 - 5.06e4T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 - 7.95e4T^{2} \)
47 \( 1 + 545. iT - 1.03e5T^{2} \)
53 \( 1 + 621. iT - 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 + 358T + 2.26e5T^{2} \)
67 \( 1 - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 3.89e5T^{2} \)
79 \( 1 - 304T + 4.93e5T^{2} \)
83 \( 1 + 1.27e3iT - 5.71e5T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 - 9.12e5T^{2} \)
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

−9.927689936219233437506608675225, −8.998185146927657392335919806678, −7.989683729233355433204870848668, −7.04781669709037726726728498242, −6.45530629260957403146053034975, −5.30979528778031898348420606998, −4.19363645949184014798872271046, −3.04180068758289183849842412893, −2.11722795354408971568822150827, −0.27542861093094999649253207011, 1.20325588315976247504360493950, 2.37582838891254151706318909644, 4.04364930900492334907922734972, 4.55754405231325323317235809012, 5.90725640786330159688938102918, 6.47143845566010955488061397665, 7.968194464863354662802127303707, 8.454239045312906815053421760119, 9.207350401499362880034690101466, 10.37790573963972832287342500739

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