Properties

Label 2-4600-1.1-c1-0-101
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  + 2.29·3-s + 1.61·7-s + 2.26·9-s − 2.79·11-s − 4.88·13-s − 3.54·17-s − 2.79·19-s + 3.71·21-s + 23-s − 1.67·27-s − 3.36·29-s − 4.84·31-s − 6.41·33-s − 3.87·37-s − 11.2·39-s − 0.327·41-s + 5.38·43-s − 0.0121·47-s − 4.38·49-s − 8.13·51-s + 0.866·53-s − 6.41·57-s + 2.96·59-s − 7.29·61-s + 3.67·63-s − 6.94·67-s + 2.29·69-s + ⋯
L(s)  = 1  + 1.32·3-s + 0.611·7-s + 0.756·9-s − 0.841·11-s − 1.35·13-s − 0.859·17-s − 0.640·19-s + 0.810·21-s + 0.208·23-s − 0.322·27-s − 0.624·29-s − 0.870·31-s − 1.11·33-s − 0.636·37-s − 1.79·39-s − 0.0512·41-s + 0.820·43-s − 0.00176·47-s − 0.626·49-s − 1.13·51-s + 0.118·53-s − 0.849·57-s + 0.386·59-s − 0.934·61-s + 0.462·63-s − 0.847·67-s + 0.276·69-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 \( 1 - 2.29T + 3T^{2} \)
7 \( 1 - 1.61T + 7T^{2} \)
11 \( 1 + 2.79T + 11T^{2} \)
13 \( 1 + 4.88T + 13T^{2} \)
17 \( 1 + 3.54T + 17T^{2} \)
19 \( 1 + 2.79T + 19T^{2} \)
29 \( 1 + 3.36T + 29T^{2} \)
31 \( 1 + 4.84T + 31T^{2} \)
37 \( 1 + 3.87T + 37T^{2} \)
41 \( 1 + 0.327T + 41T^{2} \)
43 \( 1 - 5.38T + 43T^{2} \)
47 \( 1 + 0.0121T + 47T^{2} \)
53 \( 1 - 0.866T + 53T^{2} \)
59 \( 1 - 2.96T + 59T^{2} \)
61 \( 1 + 7.29T + 61T^{2} \)
67 \( 1 + 6.94T + 67T^{2} \)
71 \( 1 + 2.16T + 71T^{2} \)
73 \( 1 - 6.02T + 73T^{2} \)
79 \( 1 - 6.50T + 79T^{2} \)
83 \( 1 - 7.88T + 83T^{2} \)
89 \( 1 + 9.71T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−7.898544425903240918946716264020, −7.53172706051839284473593738712, −6.77463187033369825707347228627, −5.61504317315141930715135413141, −4.86691070869487248317629918308, −4.15459887210032792754795048357, −3.17068458023189114495646357554, −2.37484328509261378529483379612, −1.86413073333777906906533203244, 0, 1.86413073333777906906533203244, 2.37484328509261378529483379612, 3.17068458023189114495646357554, 4.15459887210032792754795048357, 4.86691070869487248317629918308, 5.61504317315141930715135413141, 6.77463187033369825707347228627, 7.53172706051839284473593738712, 7.898544425903240918946716264020

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