Properties

Label 2-4608-8.5-c1-0-66
Degree $2$
Conductor $4608$
Sign $i$
Analytic cond. $36.7950$
Root an. cond. $6.06589$
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  − 0.585i·5-s + 3.41·7-s + 2i·11-s − 2.82i·13-s − 3.65·17-s − 5.65i·19-s − 1.17·23-s + 4.65·25-s + 0.585i·29-s − 4.58·31-s − 2i·35-s − 9.65i·37-s − 11.6·41-s − 1.65i·43-s + 12.4·47-s + ⋯
L(s)  = 1  − 0.261i·5-s + 1.29·7-s + 0.603i·11-s − 0.784i·13-s − 0.886·17-s − 1.29i·19-s − 0.244·23-s + 0.931·25-s + 0.108i·29-s − 0.823·31-s − 0.338i·35-s − 1.58i·37-s − 1.82·41-s − 0.252i·43-s + 1.82·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $i$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.782518031\)
\(L(\frac12)\) \(\approx\) \(1.782518031\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + 0.585iT - 5T^{2} \)
7 \( 1 - 3.41T + 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + 2.82iT - 13T^{2} \)
17 \( 1 + 3.65T + 17T^{2} \)
19 \( 1 + 5.65iT - 19T^{2} \)
23 \( 1 + 1.17T + 23T^{2} \)
29 \( 1 - 0.585iT - 29T^{2} \)
31 \( 1 + 4.58T + 31T^{2} \)
37 \( 1 + 9.65iT - 37T^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 + 1.65iT - 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 + 11.8iT - 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 - 9.65iT - 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 9.17T + 71T^{2} \)
73 \( 1 - 1.65T + 73T^{2} \)
79 \( 1 + 5.75T + 79T^{2} \)
83 \( 1 + 9.31iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 13.3T + 97T^{2} \)
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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

−8.211756397241745031837524435246, −7.33601617005279020687428447417, −6.94211388283690816633526906130, −5.78735155800725253088443953219, −5.00271437312430610242654086205, −4.64417730919133032585895679480, −3.64758591128935322869634687045, −2.46868646195459125563240609431, −1.74728279268091215083279673267, −0.48957208791820352836539566835, 1.28978898462738994576129746431, 2.03641760690677580982251372192, 3.12153925006333445692482880975, 4.10679164714186094572481860631, 4.72336500395271135453726009694, 5.56477740766218891017586581507, 6.32911848509382014739485125078, 7.09377387287419441515180803916, 7.82040008073349870024172524330, 8.577028004285053320171807620499

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