Properties

Label 2-4600-1.1-c1-0-99
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  + 2.95·3-s − 3.95·7-s + 5.74·9-s − 0.957·11-s + 2.74·13-s − 5.74·17-s − 6.74·19-s − 11.7·21-s − 23-s + 8.12·27-s + 5.21·29-s − 5.95·31-s − 2.83·33-s − 9.12·37-s + 8.12·39-s + 0.252·41-s − 8·43-s − 5.49·47-s + 8.66·49-s − 16.9·51-s + 7.12·53-s − 19.9·57-s − 4.78·59-s + 12.4·61-s − 22.7·63-s − 9.12·67-s − 2.95·69-s + ⋯
L(s)  = 1  + 1.70·3-s − 1.49·7-s + 1.91·9-s − 0.288·11-s + 0.761·13-s − 1.39·17-s − 1.54·19-s − 2.55·21-s − 0.208·23-s + 1.56·27-s + 0.967·29-s − 1.07·31-s − 0.493·33-s − 1.50·37-s + 1.30·39-s + 0.0394·41-s − 1.21·43-s − 0.801·47-s + 1.23·49-s − 2.38·51-s + 0.978·53-s − 2.64·57-s − 0.623·59-s + 1.59·61-s − 2.86·63-s − 1.11·67-s − 0.356·69-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 \( 1 - 2.95T + 3T^{2} \)
7 \( 1 + 3.95T + 7T^{2} \)
11 \( 1 + 0.957T + 11T^{2} \)
13 \( 1 - 2.74T + 13T^{2} \)
17 \( 1 + 5.74T + 17T^{2} \)
19 \( 1 + 6.74T + 19T^{2} \)
29 \( 1 - 5.21T + 29T^{2} \)
31 \( 1 + 5.95T + 31T^{2} \)
37 \( 1 + 9.12T + 37T^{2} \)
41 \( 1 - 0.252T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 5.49T + 47T^{2} \)
53 \( 1 - 7.12T + 53T^{2} \)
59 \( 1 + 4.78T + 59T^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 + 9.12T + 67T^{2} \)
71 \( 1 - 1.66T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 + 0.704T + 83T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−8.297338692479269639304545564627, −7.16064094146726709545067251720, −6.71658919030604955381364347196, −6.02654374442356144941112713072, −4.68813612639561842765545932106, −3.84352366124386697701380113780, −3.35330869837054529412550564426, −2.51563998092707630371367353270, −1.80847215035675707208142687497, 0, 1.80847215035675707208142687497, 2.51563998092707630371367353270, 3.35330869837054529412550564426, 3.84352366124386697701380113780, 4.68813612639561842765545932106, 6.02654374442356144941112713072, 6.71658919030604955381364347196, 7.16064094146726709545067251720, 8.297338692479269639304545564627

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