Properties

Label 2-4600-1.1-c1-0-70
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·7-s − 2·9-s − 13-s + 4·17-s − 4·19-s − 2·21-s − 23-s + 5·27-s − 3·29-s − 31-s + 8·37-s + 39-s − 5·41-s + 6·43-s − 9·47-s − 3·49-s − 4·51-s − 2·53-s + 4·57-s − 4·63-s − 4·67-s + 69-s + 3·71-s − 7·73-s + 4·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s − 2/3·9-s − 0.277·13-s + 0.970·17-s − 0.917·19-s − 0.436·21-s − 0.208·23-s + 0.962·27-s − 0.557·29-s − 0.179·31-s + 1.31·37-s + 0.160·39-s − 0.780·41-s + 0.914·43-s − 1.31·47-s − 3/7·49-s − 0.560·51-s − 0.274·53-s + 0.529·57-s − 0.503·63-s − 0.488·67-s + 0.120·69-s + 0.356·71-s − 0.819·73-s + 0.450·79-s + 1/9·81-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: \(1\)
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 \( 1 + T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.026104839206072484718820015734, −7.28357641965222627403634626356, −6.33573760181621128718429717667, −5.76751732391269470885301891120, −5.04777616072826196553892724666, −4.37223386482384364604870545239, −3.33910206209539937228556444848, −2.37686439291735034085386510972, −1.31679312432706868249748039940, 0, 1.31679312432706868249748039940, 2.37686439291735034085386510972, 3.33910206209539937228556444848, 4.37223386482384364604870545239, 5.04777616072826196553892724666, 5.76751732391269470885301891120, 6.33573760181621128718429717667, 7.28357641965222627403634626356, 8.026104839206072484718820015734

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