Properties

Label 2-4600-1.1-c1-0-54
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  − 3.08·3-s + 0.555·7-s + 6.50·9-s + 4.65·11-s − 5.02·13-s − 1.32·17-s − 0.196·19-s − 1.71·21-s − 23-s − 10.8·27-s − 0.812·29-s − 2.11·31-s − 14.3·33-s + 5.64·37-s + 15.4·39-s − 4.89·41-s − 1.66·43-s + 9.89·47-s − 6.69·49-s + 4.09·51-s − 2.23·53-s + 0.605·57-s + 2.43·59-s − 5.71·61-s + 3.61·63-s − 6.91·67-s + 3.08·69-s + ⋯
L(s)  = 1  − 1.78·3-s + 0.209·7-s + 2.16·9-s + 1.40·11-s − 1.39·13-s − 0.322·17-s − 0.0450·19-s − 0.373·21-s − 0.208·23-s − 2.08·27-s − 0.150·29-s − 0.379·31-s − 2.49·33-s + 0.928·37-s + 2.47·39-s − 0.764·41-s − 0.253·43-s + 1.44·47-s − 0.955·49-s + 0.573·51-s − 0.306·53-s + 0.0802·57-s + 0.316·59-s − 0.731·61-s + 0.455·63-s − 0.845·67-s + 0.371·69-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 \( 1 + 3.08T + 3T^{2} \)
7 \( 1 - 0.555T + 7T^{2} \)
11 \( 1 - 4.65T + 11T^{2} \)
13 \( 1 + 5.02T + 13T^{2} \)
17 \( 1 + 1.32T + 17T^{2} \)
19 \( 1 + 0.196T + 19T^{2} \)
29 \( 1 + 0.812T + 29T^{2} \)
31 \( 1 + 2.11T + 31T^{2} \)
37 \( 1 - 5.64T + 37T^{2} \)
41 \( 1 + 4.89T + 41T^{2} \)
43 \( 1 + 1.66T + 43T^{2} \)
47 \( 1 - 9.89T + 47T^{2} \)
53 \( 1 + 2.23T + 53T^{2} \)
59 \( 1 - 2.43T + 59T^{2} \)
61 \( 1 + 5.71T + 61T^{2} \)
67 \( 1 + 6.91T + 67T^{2} \)
71 \( 1 - 0.0120T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 - 10.6T + 79T^{2} \)
83 \( 1 - 5.64T + 83T^{2} \)
89 \( 1 - 9.06T + 89T^{2} \)
97 \( 1 - 14.0T + 97T^{2} \)
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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

−7.58349048886454151094128065800, −7.13528182243101504792720422751, −6.32313819828053256553287621221, −5.91676086879832799720752753584, −4.87849080533823714397843837810, −4.58350036803510447691693075582, −3.61473123589324220854371874623, −2.13185409700248642782152472093, −1.13492912378552764650173446555, 0, 1.13492912378552764650173446555, 2.13185409700248642782152472093, 3.61473123589324220854371874623, 4.58350036803510447691693075582, 4.87849080533823714397843837810, 5.91676086879832799720752753584, 6.32313819828053256553287621221, 7.13528182243101504792720422751, 7.58349048886454151094128065800

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