Properties

Label 2-4600-1.1-c1-0-38
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  − 3·3-s + 2·7-s + 6·9-s + 5·13-s + 6·17-s + 6·19-s − 6·21-s − 23-s − 9·27-s + 9·29-s + 3·31-s + 8·37-s − 15·39-s + 3·41-s + 8·43-s − 7·47-s − 3·49-s − 18·51-s + 2·53-s − 18·57-s + 4·59-s − 10·61-s + 12·63-s − 8·67-s + 3·69-s + 7·71-s − 9·73-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.755·7-s + 2·9-s + 1.38·13-s + 1.45·17-s + 1.37·19-s − 1.30·21-s − 0.208·23-s − 1.73·27-s + 1.67·29-s + 0.538·31-s + 1.31·37-s − 2.40·39-s + 0.468·41-s + 1.21·43-s − 1.02·47-s − 3/7·49-s − 2.52·51-s + 0.274·53-s − 2.38·57-s + 0.520·59-s − 1.28·61-s + 1.51·63-s − 0.977·67-s + 0.361·69-s + 0.830·71-s − 1.05·73-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

\(L(1)\) \(\approx\) \(1.568026343\)
\(L(\frac12)\) \(\approx\) \(1.568026343\)
\(L(\frac{3}{2})\) 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 \( 1 + p T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−7.997402643331904740009462370615, −7.67899822443948661188579997050, −6.58275003853327185223548469271, −6.06483814765580990692180218296, −5.44663656641282997507750663047, −4.81358070969442574628445235427, −4.04667320938686712108909524918, −2.98213621017603004203348344774, −1.30055558518431425089443698022, −0.950133350562753842827187537167, 0.950133350562753842827187537167, 1.30055558518431425089443698022, 2.98213621017603004203348344774, 4.04667320938686712108909524918, 4.81358070969442574628445235427, 5.44663656641282997507750663047, 6.06483814765580990692180218296, 6.58275003853327185223548469271, 7.67899822443948661188579997050, 7.997402643331904740009462370615

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