Properties

Label 2-4600-1.1-c1-0-96
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  + 1.78·3-s + 1.75·7-s + 0.192·9-s − 4.77·11-s + 1.72·13-s − 7.81·17-s − 2.43·19-s + 3.13·21-s − 23-s − 5.01·27-s + 7.86·29-s + 6.14·31-s − 8.53·33-s − 6.83·37-s + 3.09·39-s − 2.50·41-s − 3.26·43-s + 8.46·47-s − 3.91·49-s − 13.9·51-s − 2.76·53-s − 4.35·57-s + 1.91·59-s − 3.50·61-s + 0.337·63-s − 12.7·67-s − 1.78·69-s + ⋯
L(s)  = 1  + 1.03·3-s + 0.663·7-s + 0.0640·9-s − 1.43·11-s + 0.479·13-s − 1.89·17-s − 0.559·19-s + 0.684·21-s − 0.208·23-s − 0.965·27-s + 1.46·29-s + 1.10·31-s − 1.48·33-s − 1.12·37-s + 0.494·39-s − 0.391·41-s − 0.498·43-s + 1.23·47-s − 0.559·49-s − 1.95·51-s − 0.379·53-s − 0.577·57-s + 0.249·59-s − 0.448·61-s + 0.0424·63-s − 1.55·67-s − 0.215·69-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 \( 1 - 1.78T + 3T^{2} \)
7 \( 1 - 1.75T + 7T^{2} \)
11 \( 1 + 4.77T + 11T^{2} \)
13 \( 1 - 1.72T + 13T^{2} \)
17 \( 1 + 7.81T + 17T^{2} \)
19 \( 1 + 2.43T + 19T^{2} \)
29 \( 1 - 7.86T + 29T^{2} \)
31 \( 1 - 6.14T + 31T^{2} \)
37 \( 1 + 6.83T + 37T^{2} \)
41 \( 1 + 2.50T + 41T^{2} \)
43 \( 1 + 3.26T + 43T^{2} \)
47 \( 1 - 8.46T + 47T^{2} \)
53 \( 1 + 2.76T + 53T^{2} \)
59 \( 1 - 1.91T + 59T^{2} \)
61 \( 1 + 3.50T + 61T^{2} \)
67 \( 1 + 12.7T + 67T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 - 0.0111T + 73T^{2} \)
79 \( 1 + 16.6T + 79T^{2} \)
83 \( 1 + 2.64T + 83T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 - 15.8T + 97T^{2} \)
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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

−8.192975007727421162373406811924, −7.43252467855737903607379942607, −6.58962174514932477058685808409, −5.76680200050340035488492600572, −4.76101641547219370630311666157, −4.29749585903872053748224646288, −3.07740331263286469319310028460, −2.52562473655515477203955694368, −1.70675689675953100502713882154, 0, 1.70675689675953100502713882154, 2.52562473655515477203955694368, 3.07740331263286469319310028460, 4.29749585903872053748224646288, 4.76101641547219370630311666157, 5.76680200050340035488492600572, 6.58962174514932477058685808409, 7.43252467855737903607379942607, 8.192975007727421162373406811924

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