Properties

Label 2-4600-1.1-c1-0-102
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  + 1.58·3-s + 2.84·7-s − 0.493·9-s + 1.98·11-s − 4.69·13-s − 3.16·17-s − 6.16·19-s + 4.50·21-s − 23-s − 5.53·27-s − 6.61·29-s − 8.29·31-s + 3.14·33-s + 1.71·37-s − 7.42·39-s + 6.72·41-s + 0.177·43-s − 11.4·47-s + 1.10·49-s − 5.01·51-s − 6.18·53-s − 9.75·57-s − 7.61·59-s + 11.8·61-s − 1.40·63-s − 3.07·67-s − 1.58·69-s + ⋯
L(s)  = 1  + 0.914·3-s + 1.07·7-s − 0.164·9-s + 0.598·11-s − 1.30·13-s − 0.768·17-s − 1.41·19-s + 0.983·21-s − 0.208·23-s − 1.06·27-s − 1.22·29-s − 1.49·31-s + 0.547·33-s + 0.281·37-s − 1.18·39-s + 1.05·41-s + 0.0271·43-s − 1.67·47-s + 0.157·49-s − 0.702·51-s − 0.849·53-s − 1.29·57-s − 0.991·59-s + 1.51·61-s − 0.176·63-s − 0.375·67-s − 0.190·69-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 \( 1 - 1.58T + 3T^{2} \)
7 \( 1 - 2.84T + 7T^{2} \)
11 \( 1 - 1.98T + 11T^{2} \)
13 \( 1 + 4.69T + 13T^{2} \)
17 \( 1 + 3.16T + 17T^{2} \)
19 \( 1 + 6.16T + 19T^{2} \)
29 \( 1 + 6.61T + 29T^{2} \)
31 \( 1 + 8.29T + 31T^{2} \)
37 \( 1 - 1.71T + 37T^{2} \)
41 \( 1 - 6.72T + 41T^{2} \)
43 \( 1 - 0.177T + 43T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 + 6.18T + 53T^{2} \)
59 \( 1 + 7.61T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 + 3.07T + 67T^{2} \)
71 \( 1 + 1.96T + 71T^{2} \)
73 \( 1 + 4.94T + 73T^{2} \)
79 \( 1 - 8.53T + 79T^{2} \)
83 \( 1 + 3.49T + 83T^{2} \)
89 \( 1 - 7.07T + 89T^{2} \)
97 \( 1 - 7.00T + 97T^{2} \)
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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

−7.929549556796833816011723470393, −7.50867469461560118196308996308, −6.61437830556723633219841041686, −5.72684255678986060211968718992, −4.81606388359797338963570651609, −4.20931925617068692541836080379, −3.32010419563285426020089737470, −2.17801451802281865148522756127, −1.87274060247329682452644795993, 0, 1.87274060247329682452644795993, 2.17801451802281865148522756127, 3.32010419563285426020089737470, 4.20931925617068692541836080379, 4.81606388359797338963570651609, 5.72684255678986060211968718992, 6.61437830556723633219841041686, 7.50867469461560118196308996308, 7.929549556796833816011723470393

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