Properties

Label 2-4608-1.1-c1-0-68
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.24·11-s − 5.65·17-s − 7.41·19-s − 5·25-s + 6·41-s − 13.0·43-s − 7·49-s − 14.2·59-s + 3.89·67-s + 16.9·73-s − 10.7·83-s + 5.65·89-s − 16.9·97-s − 9.75·107-s + 18·113-s + ⋯
L(s)  = 1  + 1.88·11-s − 1.37·17-s − 1.70·19-s − 25-s + 0.937·41-s − 1.99·43-s − 49-s − 1.85·59-s + 0.476·67-s + 1.98·73-s − 1.17·83-s + 0.599·89-s − 1.72·97-s − 0.943·107-s + 1.69·113-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: \(1\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 7T^{2} \)
11 \( 1 - 6.24T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 5.65T + 17T^{2} \)
19 \( 1 + 7.41T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 13.0T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 14.2T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 3.89T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 16.9T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 - 5.65T + 89T^{2} \)
97 \( 1 + 16.9T + 97T^{2} \)
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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

−8.137710831311582676960284026210, −7.01981346526406732909773271062, −6.47273646419501755340965739900, −6.06000294258383803448846873390, −4.77263278272474420224490343176, −4.19476945821425322125102589707, −3.52911992619736380149138712677, −2.24930016470567515669027977300, −1.51801355918979701531711276996, 0, 1.51801355918979701531711276996, 2.24930016470567515669027977300, 3.52911992619736380149138712677, 4.19476945821425322125102589707, 4.77263278272474420224490343176, 6.06000294258383803448846873390, 6.47273646419501755340965739900, 7.01981346526406732909773271062, 8.137710831311582676960284026210

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