Properties

Label 2-4600-1.1-c1-0-5
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.56·3-s − 0.561·9-s − 3.12·11-s − 0.438·13-s − 5.12·17-s − 3.12·19-s + 23-s + 5.56·27-s + 3.56·29-s − 2.43·31-s + 4.87·33-s − 8.24·37-s + 0.684·39-s − 9.80·41-s + 8·43-s + 0.684·47-s − 7·49-s + 8·51-s − 2·53-s + 4.87·57-s + 10.2·59-s − 4.24·61-s − 3.12·67-s − 1.56·69-s + 13.5·71-s + 14.6·73-s + 3.12·79-s + ⋯
L(s)  = 1  − 0.901·3-s − 0.187·9-s − 0.941·11-s − 0.121·13-s − 1.24·17-s − 0.716·19-s + 0.208·23-s + 1.07·27-s + 0.661·29-s − 0.437·31-s + 0.848·33-s − 1.35·37-s + 0.109·39-s − 1.53·41-s + 1.21·43-s + 0.0998·47-s − 49-s + 1.12·51-s − 0.274·53-s + 0.645·57-s + 1.33·59-s − 0.543·61-s − 0.381·67-s − 0.187·69-s + 1.60·71-s + 1.71·73-s + 0.351·79-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: \(0\)
Selberg data: \((2,\ 4600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6243922875\)
\(L(\frac12)\) \(\approx\) \(0.6243922875\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.56T + 3T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 3.12T + 11T^{2} \)
13 \( 1 + 0.438T + 13T^{2} \)
17 \( 1 + 5.12T + 17T^{2} \)
19 \( 1 + 3.12T + 19T^{2} \)
29 \( 1 - 3.56T + 29T^{2} \)
31 \( 1 + 2.43T + 31T^{2} \)
37 \( 1 + 8.24T + 37T^{2} \)
41 \( 1 + 9.80T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 0.684T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 + 4.24T + 61T^{2} \)
67 \( 1 + 3.12T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 - 14.6T + 73T^{2} \)
79 \( 1 - 3.12T + 79T^{2} \)
83 \( 1 + 14.2T + 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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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

−8.459315617046651073932519656180, −7.50940527140274791465455324565, −6.66152523171619072891076458678, −6.24398949123731821073498930327, −5.17774338010131377346750686654, −4.95684830201100696518791760752, −3.87279828490676008199182221474, −2.80778781793794385017375465096, −1.94983174031447491261170566818, −0.43718609551609084021564637761, 0.43718609551609084021564637761, 1.94983174031447491261170566818, 2.80778781793794385017375465096, 3.87279828490676008199182221474, 4.95684830201100696518791760752, 5.17774338010131377346750686654, 6.24398949123731821073498930327, 6.66152523171619072891076458678, 7.50940527140274791465455324565, 8.459315617046651073932519656180

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