Properties

Label 2-4600-1.1-c1-0-39
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-s + 4·7-s − 2·9-s + 3·11-s − 2·13-s − 17-s − 19-s + 4·21-s + 23-s − 5·27-s + 8·31-s + 3·33-s − 2·37-s − 2·39-s + 41-s + 12·43-s + 6·47-s + 9·49-s − 51-s − 4·53-s − 57-s + 12·59-s − 8·63-s + 13·67-s + 69-s + 12·71-s − 17·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.51·7-s − 2/3·9-s + 0.904·11-s − 0.554·13-s − 0.242·17-s − 0.229·19-s + 0.872·21-s + 0.208·23-s − 0.962·27-s + 1.43·31-s + 0.522·33-s − 0.328·37-s − 0.320·39-s + 0.156·41-s + 1.82·43-s + 0.875·47-s + 9/7·49-s − 0.140·51-s − 0.549·53-s − 0.132·57-s + 1.56·59-s − 1.00·63-s + 1.58·67-s + 0.120·69-s + 1.42·71-s − 1.98·73-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

\(L(1)\) \(\approx\) \(2.879311261\)
\(L(\frac12)\) \(\approx\) \(2.879311261\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 17 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 5 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.390902451562958491957440349762, −7.73570175095482955640696517468, −7.03454162395028209903297604348, −6.09771145409907104404464047508, −5.30866601482743800419366313081, −4.52378274870349201159710526000, −3.87866168934077473105692873502, −2.71690637658407031705062332425, −2.05547272414673488375387186861, −0.954736524589207365541990687901, 0.954736524589207365541990687901, 2.05547272414673488375387186861, 2.71690637658407031705062332425, 3.87866168934077473105692873502, 4.52378274870349201159710526000, 5.30866601482743800419366313081, 6.09771145409907104404464047508, 7.03454162395028209903297604348, 7.73570175095482955640696517468, 8.390902451562958491957440349762

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