Properties

Label 2-4600-1.1-c1-0-94
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  + 2.21·3-s − 2.22·7-s + 1.88·9-s − 1.57·11-s + 3.96·13-s + 0.294·17-s − 7.76·19-s − 4.91·21-s − 23-s − 2.46·27-s − 9.29·29-s + 9.18·31-s − 3.47·33-s − 10.5·37-s + 8.75·39-s − 2.34·41-s + 6.67·43-s − 1.38·47-s − 2.04·49-s + 0.651·51-s − 11.0·53-s − 17.1·57-s − 5.09·59-s − 8.91·61-s − 4.19·63-s − 1.12·67-s − 2.21·69-s + ⋯
L(s)  = 1  + 1.27·3-s − 0.840·7-s + 0.628·9-s − 0.474·11-s + 1.09·13-s + 0.0715·17-s − 1.78·19-s − 1.07·21-s − 0.208·23-s − 0.473·27-s − 1.72·29-s + 1.65·31-s − 0.605·33-s − 1.73·37-s + 1.40·39-s − 0.365·41-s + 1.01·43-s − 0.201·47-s − 0.292·49-s + 0.0912·51-s − 1.51·53-s − 2.27·57-s − 0.663·59-s − 1.14·61-s − 0.528·63-s − 0.136·67-s − 0.266·69-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 \( 1 - 2.21T + 3T^{2} \)
7 \( 1 + 2.22T + 7T^{2} \)
11 \( 1 + 1.57T + 11T^{2} \)
13 \( 1 - 3.96T + 13T^{2} \)
17 \( 1 - 0.294T + 17T^{2} \)
19 \( 1 + 7.76T + 19T^{2} \)
29 \( 1 + 9.29T + 29T^{2} \)
31 \( 1 - 9.18T + 31T^{2} \)
37 \( 1 + 10.5T + 37T^{2} \)
41 \( 1 + 2.34T + 41T^{2} \)
43 \( 1 - 6.67T + 43T^{2} \)
47 \( 1 + 1.38T + 47T^{2} \)
53 \( 1 + 11.0T + 53T^{2} \)
59 \( 1 + 5.09T + 59T^{2} \)
61 \( 1 + 8.91T + 61T^{2} \)
67 \( 1 + 1.12T + 67T^{2} \)
71 \( 1 - 7.60T + 71T^{2} \)
73 \( 1 - 12.8T + 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 + 13.5T + 83T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 + 0.199T + 97T^{2} \)
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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

−8.190545391686223136816251581775, −7.40149081292383380142337297440, −6.43884072777812617560278996093, −6.02170022756131713076601414324, −4.84498876579350754775107016715, −3.83263244786778825739887429230, −3.39325117997997099877192420350, −2.50670749626134397358955452780, −1.69926867271656180868227390547, 0, 1.69926867271656180868227390547, 2.50670749626134397358955452780, 3.39325117997997099877192420350, 3.83263244786778825739887429230, 4.84498876579350754775107016715, 6.02170022756131713076601414324, 6.43884072777812617560278996093, 7.40149081292383380142337297440, 8.190545391686223136816251581775

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