Properties

Label 2-4600-1.1-c1-0-12
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.30·3-s + 2.55·7-s + 7.93·9-s − 2.72·11-s − 7.12·13-s − 0.924·17-s + 7.51·19-s − 8.43·21-s + 23-s − 16.3·27-s − 2.38·29-s + 0.866·31-s + 9.00·33-s − 0.352·37-s + 23.5·39-s + 4.34·41-s + 13.3·47-s − 0.495·49-s + 3.05·51-s − 3.99·53-s − 24.8·57-s − 3.84·59-s − 9.14·61-s + 20.2·63-s + 3.15·67-s − 3.30·69-s − 6.07·71-s + ⋯
L(s)  = 1  − 1.90·3-s + 0.963·7-s + 2.64·9-s − 0.821·11-s − 1.97·13-s − 0.224·17-s + 1.72·19-s − 1.84·21-s + 0.208·23-s − 3.13·27-s − 0.442·29-s + 0.155·31-s + 1.56·33-s − 0.0580·37-s + 3.77·39-s + 0.677·41-s + 1.94·47-s − 0.0707·49-s + 0.427·51-s − 0.548·53-s − 3.29·57-s − 0.500·59-s − 1.17·61-s + 2.54·63-s + 0.385·67-s − 0.398·69-s − 0.721·71-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: \(0\)
Selberg data: \((2,\ 4600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7602509044\)
\(L(\frac12)\) \(\approx\) \(0.7602509044\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.30T + 3T^{2} \)
7 \( 1 - 2.55T + 7T^{2} \)
11 \( 1 + 2.72T + 11T^{2} \)
13 \( 1 + 7.12T + 13T^{2} \)
17 \( 1 + 0.924T + 17T^{2} \)
19 \( 1 - 7.51T + 19T^{2} \)
29 \( 1 + 2.38T + 29T^{2} \)
31 \( 1 - 0.866T + 31T^{2} \)
37 \( 1 + 0.352T + 37T^{2} \)
41 \( 1 - 4.34T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 13.3T + 47T^{2} \)
53 \( 1 + 3.99T + 53T^{2} \)
59 \( 1 + 3.84T + 59T^{2} \)
61 \( 1 + 9.14T + 61T^{2} \)
67 \( 1 - 3.15T + 67T^{2} \)
71 \( 1 + 6.07T + 71T^{2} \)
73 \( 1 - 11.3T + 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 - 6.35T + 83T^{2} \)
89 \( 1 + 9.71T + 89T^{2} \)
97 \( 1 + 8.76T + 97T^{2} \)
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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

−7.918735026808505999172819746189, −7.41742457667359692232236448874, −6.99600012673250607163317852327, −5.85353135208493384170537816357, −5.31290102168934650715082706296, −4.88867949014580661560886250531, −4.26058248300030371067004293913, −2.75737139730763787273918467623, −1.64350337481046940627463208629, −0.54620495600278261556248022128, 0.54620495600278261556248022128, 1.64350337481046940627463208629, 2.75737139730763787273918467623, 4.26058248300030371067004293913, 4.88867949014580661560886250531, 5.31290102168934650715082706296, 5.85353135208493384170537816357, 6.99600012673250607163317852327, 7.41742457667359692232236448874, 7.918735026808505999172819746189

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