Properties

Label 2-4608-12.11-c1-0-35
Degree $2$
Conductor $4608$
Sign $0.577 - 0.816i$
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  + 4.24i·7-s + 4·11-s + 6·13-s + 4.24i·17-s + 2.82i·19-s + 6·23-s + 5·25-s − 8.48i·29-s − 4.24i·31-s + 6·37-s − 1.41i·41-s + 2.82i·43-s − 6·47-s − 10.9·49-s − 8.48i·53-s + ⋯
L(s)  = 1  + 1.60i·7-s + 1.20·11-s + 1.66·13-s + 1.02i·17-s + 0.648i·19-s + 1.25·23-s + 25-s − 1.57i·29-s − 0.762i·31-s + 0.986·37-s − 0.220i·41-s + 0.431i·43-s − 0.875·47-s − 1.57·49-s − 1.16i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (4607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ 0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.526885928\)
\(L(\frac12)\) \(\approx\) \(2.526885928\)
\(L(\frac{3}{2})\) 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 \( 1 - 5T^{2} \)
7 \( 1 - 4.24iT - 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 - 4.24iT - 17T^{2} \)
19 \( 1 - 2.82iT - 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + 8.48iT - 29T^{2} \)
31 \( 1 + 4.24iT - 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 1.41iT - 41T^{2} \)
43 \( 1 - 2.82iT - 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 8.48iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 - 11.3iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 4.24iT - 79T^{2} \)
83 \( 1 - 16T + 83T^{2} \)
89 \( 1 + 12.7iT - 89T^{2} \)
97 \( 1 + 12T + 97T^{2} \)
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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

−8.484015295899434602061221337538, −7.989651474281278799881134277778, −6.72694023525528348780547049187, −6.05814012643584915620117160283, −5.85585439090744997395176792100, −4.68524519266277870022788608504, −3.83694027102871392400673482441, −3.08029807235012442454414782029, −2.01211967997663225720787154190, −1.14113503801256666518497202426, 0.939108887495370187249818808483, 1.30213643363648727244230431575, 3.07887638483274040555976741703, 3.56895280629375255955337280643, 4.47904134786138972421143943427, 5.04714353377098611534712036165, 6.31789516549105829615505192077, 6.81820840865345812992420634453, 7.26526592824132413419224349051, 8.223543303797020442650510989758

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