Properties

Label 2-4600-1.1-c1-0-77
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  + 0.356·3-s − 2.46·7-s − 2.87·9-s + 1.61·11-s + 2.62·13-s + 2.58·17-s + 4.02·19-s − 0.878·21-s − 23-s − 2.09·27-s − 7.08·29-s − 4.58·31-s + 0.575·33-s − 2.96·37-s + 0.935·39-s − 5.71·41-s + 2.30·43-s − 6.88·47-s − 0.929·49-s + 0.922·51-s + 6.76·53-s + 1.43·57-s + 2.53·59-s + 9.25·61-s + 7.07·63-s + 15.7·67-s − 0.356·69-s + ⋯
L(s)  = 1  + 0.205·3-s − 0.931·7-s − 0.957·9-s + 0.487·11-s + 0.728·13-s + 0.627·17-s + 0.923·19-s − 0.191·21-s − 0.208·23-s − 0.402·27-s − 1.31·29-s − 0.823·31-s + 0.100·33-s − 0.486·37-s + 0.149·39-s − 0.891·41-s + 0.351·43-s − 1.00·47-s − 0.132·49-s + 0.129·51-s + 0.929·53-s + 0.190·57-s + 0.329·59-s + 1.18·61-s + 0.891·63-s + 1.92·67-s − 0.0429·69-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 \( 1 - 0.356T + 3T^{2} \)
7 \( 1 + 2.46T + 7T^{2} \)
11 \( 1 - 1.61T + 11T^{2} \)
13 \( 1 - 2.62T + 13T^{2} \)
17 \( 1 - 2.58T + 17T^{2} \)
19 \( 1 - 4.02T + 19T^{2} \)
29 \( 1 + 7.08T + 29T^{2} \)
31 \( 1 + 4.58T + 31T^{2} \)
37 \( 1 + 2.96T + 37T^{2} \)
41 \( 1 + 5.71T + 41T^{2} \)
43 \( 1 - 2.30T + 43T^{2} \)
47 \( 1 + 6.88T + 47T^{2} \)
53 \( 1 - 6.76T + 53T^{2} \)
59 \( 1 - 2.53T + 59T^{2} \)
61 \( 1 - 9.25T + 61T^{2} \)
67 \( 1 - 15.7T + 67T^{2} \)
71 \( 1 + 5.25T + 71T^{2} \)
73 \( 1 + 6.03T + 73T^{2} \)
79 \( 1 + 1.35T + 79T^{2} \)
83 \( 1 - 8.59T + 83T^{2} \)
89 \( 1 + 8.84T + 89T^{2} \)
97 \( 1 + 8.01T + 97T^{2} \)
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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

−8.069752169682404660577578234319, −7.16811823927347339548006358213, −6.53090523905569060747220303009, −5.68109020117942368697194335742, −5.27882286286787001564779537593, −3.75071408261960247271556006558, −3.53639025723533032283837588392, −2.56013718914314383544108047239, −1.36886688188199271753749348557, 0, 1.36886688188199271753749348557, 2.56013718914314383544108047239, 3.53639025723533032283837588392, 3.75071408261960247271556006558, 5.27882286286787001564779537593, 5.68109020117942368697194335742, 6.53090523905569060747220303009, 7.16811823927347339548006358213, 8.069752169682404660577578234319

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