Properties

Label 2-4600-1.1-c1-0-84
Degree $2$
Conductor $4600$
Sign $-1$
Analytic cond. $36.7311$
Root an. cond. $6.06062$
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  + 1.36·3-s − 3.28·7-s − 1.13·9-s + 3.49·11-s + 3.41·13-s − 7.46·17-s + 3.49·19-s − 4.48·21-s + 23-s − 5.64·27-s − 3.46·29-s + 2.01·31-s + 4.77·33-s + 0.511·37-s + 4.66·39-s − 7.07·41-s − 2.76·43-s − 0.889·47-s + 3.76·49-s − 10.1·51-s − 14.2·53-s + 4.77·57-s − 4.71·59-s + 13.4·61-s + 3.71·63-s − 2.30·67-s + 1.36·69-s + ⋯
L(s)  = 1  + 0.788·3-s − 1.24·7-s − 0.377·9-s + 1.05·11-s + 0.946·13-s − 1.80·17-s + 0.802·19-s − 0.978·21-s + 0.208·23-s − 1.08·27-s − 0.643·29-s + 0.361·31-s + 0.831·33-s + 0.0840·37-s + 0.746·39-s − 1.10·41-s − 0.421·43-s − 0.129·47-s + 0.537·49-s − 1.42·51-s − 1.95·53-s + 0.632·57-s − 0.613·59-s + 1.72·61-s + 0.468·63-s − 0.281·67-s + 0.164·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4600\)    =    \(2^{3} \cdot 5^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(36.7311\)
Root analytic conductor: \(6.06062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
23 \( 1 - T \)
good3 \( 1 - 1.36T + 3T^{2} \)
7 \( 1 + 3.28T + 7T^{2} \)
11 \( 1 - 3.49T + 11T^{2} \)
13 \( 1 - 3.41T + 13T^{2} \)
17 \( 1 + 7.46T + 17T^{2} \)
19 \( 1 - 3.49T + 19T^{2} \)
29 \( 1 + 3.46T + 29T^{2} \)
31 \( 1 - 2.01T + 31T^{2} \)
37 \( 1 - 0.511T + 37T^{2} \)
41 \( 1 + 7.07T + 41T^{2} \)
43 \( 1 + 2.76T + 43T^{2} \)
47 \( 1 + 0.889T + 47T^{2} \)
53 \( 1 + 14.2T + 53T^{2} \)
59 \( 1 + 4.71T + 59T^{2} \)
61 \( 1 - 13.4T + 61T^{2} \)
67 \( 1 + 2.30T + 67T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 - 3.70T + 73T^{2} \)
79 \( 1 + 7.97T + 79T^{2} \)
83 \( 1 - 9.42T + 83T^{2} \)
89 \( 1 - 0.801T + 89T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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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

−8.147445700184492262444871974923, −7.13966345542285764936281232709, −6.46745848775082129803443314289, −6.06499290609694168657397738106, −4.90857028334821358683371695760, −3.84185520472874140010348793103, −3.41268378351922524326574516864, −2.58542720414934539616655177691, −1.51330792393091999993441701124, 0, 1.51330792393091999993441701124, 2.58542720414934539616655177691, 3.41268378351922524326574516864, 3.84185520472874140010348793103, 4.90857028334821358683371695760, 6.06499290609694168657397738106, 6.46745848775082129803443314289, 7.13966345542285764936281232709, 8.147445700184492262444871974923

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