Properties

Label 2-4600-1.1-c1-0-100
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  + 2.98·3-s − 0.980·7-s + 5.92·9-s − 6.14·11-s − 6.37·13-s + 3.36·17-s + 1.08·19-s − 2.92·21-s − 23-s + 8.72·27-s − 0.271·29-s − 8.77·31-s − 18.3·33-s − 8.84·37-s − 19.0·39-s − 4.85·41-s + 1.87·43-s + 0.196·47-s − 6.03·49-s + 10.0·51-s − 1.93·53-s + 3.23·57-s + 13.0·59-s + 6.00·61-s − 5.80·63-s + 2.26·67-s − 2.98·69-s + ⋯
L(s)  = 1  + 1.72·3-s − 0.370·7-s + 1.97·9-s − 1.85·11-s − 1.76·13-s + 0.815·17-s + 0.248·19-s − 0.639·21-s − 0.208·23-s + 1.68·27-s − 0.0503·29-s − 1.57·31-s − 3.19·33-s − 1.45·37-s − 3.04·39-s − 0.757·41-s + 0.286·43-s + 0.0286·47-s − 0.862·49-s + 1.40·51-s − 0.265·53-s + 0.428·57-s + 1.69·59-s + 0.768·61-s − 0.731·63-s + 0.277·67-s − 0.359·69-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 \( 1 - 2.98T + 3T^{2} \)
7 \( 1 + 0.980T + 7T^{2} \)
11 \( 1 + 6.14T + 11T^{2} \)
13 \( 1 + 6.37T + 13T^{2} \)
17 \( 1 - 3.36T + 17T^{2} \)
19 \( 1 - 1.08T + 19T^{2} \)
29 \( 1 + 0.271T + 29T^{2} \)
31 \( 1 + 8.77T + 31T^{2} \)
37 \( 1 + 8.84T + 37T^{2} \)
41 \( 1 + 4.85T + 41T^{2} \)
43 \( 1 - 1.87T + 43T^{2} \)
47 \( 1 - 0.196T + 47T^{2} \)
53 \( 1 + 1.93T + 53T^{2} \)
59 \( 1 - 13.0T + 59T^{2} \)
61 \( 1 - 6.00T + 61T^{2} \)
67 \( 1 - 2.26T + 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 - 1.38T + 73T^{2} \)
79 \( 1 + 4.67T + 79T^{2} \)
83 \( 1 + 15.7T + 83T^{2} \)
89 \( 1 + 11.1T + 89T^{2} \)
97 \( 1 + 10.2T + 97T^{2} \)
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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

−8.050994759835024061210270441129, −7.29389619809046008407413804925, −7.06149305751673396970915930633, −5.45318734496672611438794545919, −5.06654602836699680195870565633, −3.91199960886521535300995804731, −3.12878812474692061998713221762, −2.56676407170049838835689753677, −1.85346072833896978509969072517, 0, 1.85346072833896978509969072517, 2.56676407170049838835689753677, 3.12878812474692061998713221762, 3.91199960886521535300995804731, 5.06654602836699680195870565633, 5.45318734496672611438794545919, 7.06149305751673396970915930633, 7.29389619809046008407413804925, 8.050994759835024061210270441129

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