Properties

Degree $2$
Conductor $4704$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2.13·5-s + 9-s + 0.298·11-s + 6.43·13-s + 2.13·15-s + 1.11·17-s + 1.01·19-s − 4.29·23-s − 0.441·25-s − 27-s − 0.422·29-s + 10.6·31-s − 0.298·33-s − 5.65·37-s − 6.43·39-s + 8.32·41-s − 7.09·43-s − 2.13·45-s − 8.11·47-s − 1.11·51-s − 3.44·53-s − 0.637·55-s − 1.01·57-s − 6.46·59-s + 5.83·61-s − 13.7·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.954·5-s + 0.333·9-s + 0.0900·11-s + 1.78·13-s + 0.551·15-s + 0.270·17-s + 0.233·19-s − 0.896·23-s − 0.0883·25-s − 0.192·27-s − 0.0784·29-s + 1.91·31-s − 0.0519·33-s − 0.929·37-s − 1.03·39-s + 1.30·41-s − 1.08·43-s − 0.318·45-s − 1.18·47-s − 0.156·51-s − 0.472·53-s − 0.0859·55-s − 0.135·57-s − 0.841·59-s + 0.747·61-s − 1.70·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4704\)    =    \(2^{5} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{4704} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4704,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.271727769\)
\(L(\frac12)\) \(\approx\) \(1.271727769\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 \)
good5 \( 1 + 2.13T + 5T^{2} \)
11 \( 1 - 0.298T + 11T^{2} \)
13 \( 1 - 6.43T + 13T^{2} \)
17 \( 1 - 1.11T + 17T^{2} \)
19 \( 1 - 1.01T + 19T^{2} \)
23 \( 1 + 4.29T + 23T^{2} \)
29 \( 1 + 0.422T + 29T^{2} \)
31 \( 1 - 10.6T + 31T^{2} \)
37 \( 1 + 5.65T + 37T^{2} \)
41 \( 1 - 8.32T + 41T^{2} \)
43 \( 1 + 7.09T + 43T^{2} \)
47 \( 1 + 8.11T + 47T^{2} \)
53 \( 1 + 3.44T + 53T^{2} \)
59 \( 1 + 6.46T + 59T^{2} \)
61 \( 1 - 5.83T + 61T^{2} \)
67 \( 1 - 5.05T + 67T^{2} \)
71 \( 1 - 1.35T + 71T^{2} \)
73 \( 1 + 2.85T + 73T^{2} \)
79 \( 1 - 7.69T + 79T^{2} \)
83 \( 1 + 6.03T + 83T^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 + 0.512T + 97T^{2} \)
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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

−8.125092688265552030272501654239, −7.78310985418920701307502774168, −6.65193498131653370866992646620, −6.25302954082153309549933403993, −5.40887878256111534441085267908, −4.47509904962284871644111927011, −3.82962782758116413342268028909, −3.14452394565212414969609438568, −1.68927087074838855042359134166, −0.66824442193925949920340621981, 0.66824442193925949920340621981, 1.68927087074838855042359134166, 3.14452394565212414969609438568, 3.82962782758116413342268028909, 4.47509904962284871644111927011, 5.40887878256111534441085267908, 6.25302954082153309549933403993, 6.65193498131653370866992646620, 7.78310985418920701307502774168, 8.125092688265552030272501654239

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