Properties

Label 2-4608-1.1-c1-0-15
Degree $2$
Conductor $4608$
Sign $1$
Analytic cond. $36.7950$
Root an. cond. $6.06589$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·5-s − 4.24·7-s + 6·11-s + 5.65·13-s + 6·17-s − 4·19-s − 2.82·23-s − 2.99·25-s − 1.41·29-s − 1.41·31-s + 6·35-s − 8.48·37-s + 2·41-s − 2.82·47-s + 10.9·49-s + 9.89·53-s − 8.48·55-s + 4·59-s − 8.48·61-s − 8.00·65-s − 8·67-s + 2.82·71-s + 8·73-s − 25.4·77-s + 12.7·79-s + 2·83-s − 8.48·85-s + ⋯
L(s)  = 1  − 0.632·5-s − 1.60·7-s + 1.80·11-s + 1.56·13-s + 1.45·17-s − 0.917·19-s − 0.589·23-s − 0.599·25-s − 0.262·29-s − 0.254·31-s + 1.01·35-s − 1.39·37-s + 0.312·41-s − 0.412·47-s + 1.57·49-s + 1.35·53-s − 1.14·55-s + 0.520·59-s − 1.08·61-s − 0.992·65-s − 0.977·67-s + 0.335·71-s + 0.936·73-s − 2.90·77-s + 1.43·79-s + 0.219·83-s − 0.920·85-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.500546398\)
\(L(\frac12)\) \(\approx\) \(1.500546398\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.41T + 5T^{2} \)
7 \( 1 + 4.24T + 7T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 + 1.41T + 29T^{2} \)
31 \( 1 + 1.41T + 31T^{2} \)
37 \( 1 + 8.48T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 - 9.89T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 8.48T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 - 2.82T + 71T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−8.463361250012052771006483277465, −7.52996212629538001487316689792, −6.72033796419035218853303475002, −6.21772600845806008953235516785, −5.69167012176414116098968066887, −4.22245783298449816629213740264, −3.57309927170967201679170006179, −3.41450649520252303565876497822, −1.80112040801481364480973890168, −0.69557844073682589426892857889, 0.69557844073682589426892857889, 1.80112040801481364480973890168, 3.41450649520252303565876497822, 3.57309927170967201679170006179, 4.22245783298449816629213740264, 5.69167012176414116098968066887, 6.21772600845806008953235516785, 6.72033796419035218853303475002, 7.52996212629538001487316689792, 8.463361250012052771006483277465

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