Properties

Label 2-720-12.11-c1-0-1
Degree $2$
Conductor $720$
Sign $-0.0917 - 0.995i$
Analytic cond. $5.74922$
Root an. cond. $2.39775$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·5-s + 2.44i·7-s − 1.01·11-s + 2.24·13-s + 4.89i·19-s − 8.36·23-s − 25-s + 6i·29-s + 8.36i·31-s − 2.44·35-s + 6.24·37-s − 4.24i·41-s − 2.02i·43-s − 2.02·47-s + 1.00·49-s + ⋯
L(s)  = 1  + 0.447i·5-s + 0.925i·7-s − 0.305·11-s + 0.621·13-s + 1.12i·19-s − 1.74·23-s − 0.200·25-s + 1.11i·29-s + 1.50i·31-s − 0.414·35-s + 1.02·37-s − 0.662i·41-s − 0.309i·43-s − 0.295·47-s + 0.142·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0917 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0917 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.0917 - 0.995i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.0917 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.850945 + 0.932956i\)
\(L(\frac12)\) \(\approx\) \(0.850945 + 0.932956i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good7 \( 1 - 2.44iT - 7T^{2} \)
11 \( 1 + 1.01T + 11T^{2} \)
13 \( 1 - 2.24T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 4.89iT - 19T^{2} \)
23 \( 1 + 8.36T + 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 - 8.36iT - 31T^{2} \)
37 \( 1 - 6.24T + 37T^{2} \)
41 \( 1 + 4.24iT - 41T^{2} \)
43 \( 1 + 2.02iT - 43T^{2} \)
47 \( 1 + 2.02T + 47T^{2} \)
53 \( 1 - 8.48iT - 53T^{2} \)
59 \( 1 + 1.01T + 59T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 - 6.92iT - 67T^{2} \)
71 \( 1 - 16.7T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 + 1.43iT - 79T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 + 16.2iT - 89T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−10.54197612224808360574611896641, −9.911311938015333752315571841586, −8.791957865936912851337355598971, −8.189649294446659914128214192050, −7.13810095398466744309977725341, −6.06348527074510286696924795489, −5.48293923958868651527858830834, −4.08418608697107236636452735569, −3.01337219456184029619079923701, −1.79513214737889439934712424815, 0.65770743660628411228719284872, 2.28683391269252970577620295552, 3.82357996446729300782971816846, 4.52342692655964692987277173120, 5.75450749876556979284980092412, 6.62563093544108745472376876398, 7.76389022954765956526325611459, 8.271679359907809544547386422848, 9.512014642829879749954312179024, 10.04490410692576761671479511917

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