Properties

Label 2-4600-1.1-c1-0-44
Degree $2$
Conductor $4600$
Sign $1$
Analytic cond. $36.7311$
Root an. cond. $6.06062$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·7-s + 9-s + 5·11-s + 5·13-s + 4·17-s + 19-s − 6·21-s + 23-s + 4·27-s + 9·29-s − 2·31-s − 10·33-s + 2·37-s − 10·39-s + 3·41-s + 7·43-s + 12·47-s + 2·49-s − 8·51-s − 12·53-s − 2·57-s − 6·59-s − 10·61-s + 3·63-s + 8·67-s − 2·69-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.13·7-s + 1/3·9-s + 1.50·11-s + 1.38·13-s + 0.970·17-s + 0.229·19-s − 1.30·21-s + 0.208·23-s + 0.769·27-s + 1.67·29-s − 0.359·31-s − 1.74·33-s + 0.328·37-s − 1.60·39-s + 0.468·41-s + 1.06·43-s + 1.75·47-s + 2/7·49-s − 1.12·51-s − 1.64·53-s − 0.264·57-s − 0.781·59-s − 1.28·61-s + 0.377·63-s + 0.977·67-s − 0.240·69-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: yes
Arithmetic: yes
Character: $\chi_{4600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.990129452\)
\(L(\frac12)\) \(\approx\) \(1.990129452\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−8.350321369612509992874025462338, −7.56435142562007891775260409519, −6.65297251918000349456289919038, −6.05958714158083343162708402996, −5.53580624472769633330997403662, −4.61212767822177650453600644877, −4.04209567534456845219323238659, −2.97072320071717046138772470200, −1.40081065739806276247046171389, −1.01560926897813459933367309440, 1.01560926897813459933367309440, 1.40081065739806276247046171389, 2.97072320071717046138772470200, 4.04209567534456845219323238659, 4.61212767822177650453600644877, 5.53580624472769633330997403662, 6.05958714158083343162708402996, 6.65297251918000349456289919038, 7.56435142562007891775260409519, 8.350321369612509992874025462338

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