Properties

Label 2-4704-1.1-c1-0-65
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 + 2·5-s + 9-s − 4·11-s + 6·13-s − 2·15-s + 2·17-s − 4·19-s − 4·23-s − 25-s − 27-s − 2·29-s − 8·31-s + 4·33-s − 10·37-s − 6·39-s + 2·41-s + 8·43-s + 2·45-s − 2·51-s − 10·53-s − 8·55-s + 4·57-s + 12·59-s − 10·61-s + 12·65-s − 8·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.20·11-s + 1.66·13-s − 0.516·15-s + 0.485·17-s − 0.917·19-s − 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.696·33-s − 1.64·37-s − 0.960·39-s + 0.312·41-s + 1.21·43-s + 0.298·45-s − 0.280·51-s − 1.37·53-s − 1.07·55-s + 0.529·57-s + 1.56·59-s − 1.28·61-s + 1.48·65-s − 0.977·67-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: $\chi_{4704} (1, \cdot )$
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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.964362844350822214234423473291, −7.15677397132649365507162382918, −6.22161239173060229293717234804, −5.77693189150574398015495552381, −5.27709472311678142265210578730, −4.18545241888231423979241741982, −3.40372957038058730873728723146, −2.20870698925320336248475894717, −1.46937648965572003683650522345, 0, 1.46937648965572003683650522345, 2.20870698925320336248475894717, 3.40372957038058730873728723146, 4.18545241888231423979241741982, 5.27709472311678142265210578730, 5.77693189150574398015495552381, 6.22161239173060229293717234804, 7.15677397132649365507162382918, 7.964362844350822214234423473291

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