Properties

Degree $2$
Conductor $720$
Sign $-i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.23i·5-s + 4.47i·17-s + 4·19-s + 8.94i·23-s − 5.00·25-s − 8·31-s + 8.94i·47-s + 7·49-s + 4.47i·53-s + 2·61-s + 16·79-s − 17.8i·83-s − 10.0·85-s + 8.94i·95-s − 17.8i·107-s + ⋯
L(s)  = 1  + 0.999i·5-s + 1.08i·17-s + 0.917·19-s + 1.86i·23-s − 1.00·25-s − 1.43·31-s + 1.30i·47-s + 49-s + 0.614i·53-s + 0.256·61-s + 1.80·79-s − 1.96i·83-s − 1.08·85-s + 0.917i·95-s − 1.72i·107-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-i$
Motivic weight: \(1\)
Character: $\chi_{720} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.938328 + 0.938328i\)
\(L(\frac12)\) \(\approx\) \(0.938328 + 0.938328i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - 2.23iT \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 4.47iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 8.94iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 8.94iT - 47T^{2} \)
53 \( 1 - 4.47iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 + 17.8iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.69240128540947547888921042899, −9.809889049743542898033321700471, −9.052279995459641796793851349931, −7.76945700073456537490153964855, −7.28839388055053732670904875648, −6.16109111081262327179434600241, −5.42347997177186152207552171764, −3.93389652753075287805071163372, −3.13257995320767604323059817500, −1.73149626659091747084411437283, 0.70582787609937990066954065305, 2.29395385364598507054374219129, 3.72273210085640936662163350749, 4.84193456827757688845139374979, 5.49565879879442986829035508867, 6.72879555656138451432287786711, 7.65105595114249053970818004956, 8.601726817612098631243085833497, 9.251801175202539443198554796721, 10.07802825330082104845248678599

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