Properties

Label 2-4600-1.1-c1-0-60
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  − 2.49·3-s + 2.92·7-s + 3.24·9-s − 4.10·11-s − 0.0122·13-s + 0.155·17-s + 4.32·19-s − 7.31·21-s − 23-s − 0.616·27-s − 6.79·29-s + 2.20·31-s + 10.2·33-s − 4.60·37-s + 0.0307·39-s − 7.67·41-s + 8.38·43-s + 6.38·47-s + 1.57·49-s − 0.388·51-s − 7.80·53-s − 10.8·57-s − 14.1·59-s + 7.05·61-s + 9.50·63-s + 7.31·67-s + 2.49·69-s + ⋯
L(s)  = 1  − 1.44·3-s + 1.10·7-s + 1.08·9-s − 1.23·11-s − 0.00341·13-s + 0.0376·17-s + 0.992·19-s − 1.59·21-s − 0.208·23-s − 0.118·27-s − 1.26·29-s + 0.396·31-s + 1.78·33-s − 0.757·37-s + 0.00492·39-s − 1.19·41-s + 1.27·43-s + 0.930·47-s + 0.224·49-s − 0.0543·51-s − 1.07·53-s − 1.43·57-s − 1.83·59-s + 0.903·61-s + 1.19·63-s + 0.893·67-s + 0.300·69-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 \( 1 + 2.49T + 3T^{2} \)
7 \( 1 - 2.92T + 7T^{2} \)
11 \( 1 + 4.10T + 11T^{2} \)
13 \( 1 + 0.0122T + 13T^{2} \)
17 \( 1 - 0.155T + 17T^{2} \)
19 \( 1 - 4.32T + 19T^{2} \)
29 \( 1 + 6.79T + 29T^{2} \)
31 \( 1 - 2.20T + 31T^{2} \)
37 \( 1 + 4.60T + 37T^{2} \)
41 \( 1 + 7.67T + 41T^{2} \)
43 \( 1 - 8.38T + 43T^{2} \)
47 \( 1 - 6.38T + 47T^{2} \)
53 \( 1 + 7.80T + 53T^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 - 7.05T + 61T^{2} \)
67 \( 1 - 7.31T + 67T^{2} \)
71 \( 1 + 5.84T + 71T^{2} \)
73 \( 1 - 0.727T + 73T^{2} \)
79 \( 1 - 4.81T + 79T^{2} \)
83 \( 1 - 7.75T + 83T^{2} \)
89 \( 1 - 6.77T + 89T^{2} \)
97 \( 1 - 14.8T + 97T^{2} \)
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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

−7.73075452875673121393551331950, −7.34885545379149201911038152414, −6.32601398120873169210407823598, −5.57964673397713307349672125905, −5.12492228175319294685185108122, −4.59494854837197375359760449030, −3.43665249332236604756316086873, −2.22648227851314190355287558004, −1.18736980023198134194016845642, 0, 1.18736980023198134194016845642, 2.22648227851314190355287558004, 3.43665249332236604756316086873, 4.59494854837197375359760449030, 5.12492228175319294685185108122, 5.57964673397713307349672125905, 6.32601398120873169210407823598, 7.34885545379149201911038152414, 7.73075452875673121393551331950

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