Properties

Label 2-4704-1.1-c1-0-53
Degree $2$
Conductor $4704$
Sign $-1$
Analytic cond. $37.5616$
Root an. cond. $6.12875$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4·5-s + 9-s − 6·11-s + 5·13-s − 4·15-s + 2·17-s − 19-s + 6·23-s + 11·25-s + 27-s + 3·31-s − 6·33-s + 3·37-s + 5·39-s − 6·41-s − 5·43-s − 4·45-s + 4·47-s + 2·51-s − 6·53-s + 24·55-s − 57-s + 6·59-s − 2·61-s − 20·65-s − 7·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.78·5-s + 1/3·9-s − 1.80·11-s + 1.38·13-s − 1.03·15-s + 0.485·17-s − 0.229·19-s + 1.25·23-s + 11/5·25-s + 0.192·27-s + 0.538·31-s − 1.04·33-s + 0.493·37-s + 0.800·39-s − 0.937·41-s − 0.762·43-s − 0.596·45-s + 0.583·47-s + 0.280·51-s − 0.824·53-s + 3.23·55-s − 0.132·57-s + 0.781·59-s − 0.256·61-s − 2.48·65-s − 0.855·67-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: $-1$
Analytic conductor: \(37.5616\)
Root analytic conductor: \(6.12875\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4704,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 \)
good5 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−8.020702153597593826005621758200, −7.44518182825793648796699388529, −6.76119232242548865955576664601, −5.64083139259093461051377031785, −4.78693827245934386213661611755, −4.10379574996364177381840299897, −3.21692038150917799451949122824, −2.83482259154835414971225743890, −1.24585240373791538687525573477, 0, 1.24585240373791538687525573477, 2.83482259154835414971225743890, 3.21692038150917799451949122824, 4.10379574996364177381840299897, 4.78693827245934386213661611755, 5.64083139259093461051377031785, 6.76119232242548865955576664601, 7.44518182825793648796699388529, 8.020702153597593826005621758200

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