Properties

Label 2-8752-1.1-c1-0-199
Degree $2$
Conductor $8752$
Sign $-1$
Analytic cond. $69.8850$
Root an. cond. $8.35972$
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  − 0.544·3-s + 0.962·5-s + 3.25·7-s − 2.70·9-s + 0.883·11-s − 4.48·13-s − 0.523·15-s − 3.18·17-s − 4.12·19-s − 1.77·21-s − 3.73·23-s − 4.07·25-s + 3.10·27-s + 6.33·29-s + 8.47·31-s − 0.480·33-s + 3.13·35-s + 11.0·37-s + 2.43·39-s + 1.41·41-s + 9.31·43-s − 2.60·45-s + 5.82·47-s + 3.58·49-s + 1.73·51-s − 7.06·53-s + 0.850·55-s + ⋯
L(s)  = 1  − 0.314·3-s + 0.430·5-s + 1.22·7-s − 0.901·9-s + 0.266·11-s − 1.24·13-s − 0.135·15-s − 0.771·17-s − 0.946·19-s − 0.386·21-s − 0.779·23-s − 0.814·25-s + 0.597·27-s + 1.17·29-s + 1.52·31-s − 0.0836·33-s + 0.529·35-s + 1.81·37-s + 0.390·39-s + 0.220·41-s + 1.41·43-s − 0.388·45-s + 0.849·47-s + 0.511·49-s + 0.242·51-s − 0.970·53-s + 0.114·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8752\)    =    \(2^{4} \cdot 547\)
Sign: $-1$
Analytic conductor: \(69.8850\)
Root analytic conductor: \(8.35972\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8752,\ (\ :1/2),\ -1)\)

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 \( 1 \)
547 \( 1 - T \)
good3 \( 1 + 0.544T + 3T^{2} \)
5 \( 1 - 0.962T + 5T^{2} \)
7 \( 1 - 3.25T + 7T^{2} \)
11 \( 1 - 0.883T + 11T^{2} \)
13 \( 1 + 4.48T + 13T^{2} \)
17 \( 1 + 3.18T + 17T^{2} \)
19 \( 1 + 4.12T + 19T^{2} \)
23 \( 1 + 3.73T + 23T^{2} \)
29 \( 1 - 6.33T + 29T^{2} \)
31 \( 1 - 8.47T + 31T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
41 \( 1 - 1.41T + 41T^{2} \)
43 \( 1 - 9.31T + 43T^{2} \)
47 \( 1 - 5.82T + 47T^{2} \)
53 \( 1 + 7.06T + 53T^{2} \)
59 \( 1 + 12.9T + 59T^{2} \)
61 \( 1 + 3.04T + 61T^{2} \)
67 \( 1 - 9.15T + 67T^{2} \)
71 \( 1 - 4.57T + 71T^{2} \)
73 \( 1 + 14.4T + 73T^{2} \)
79 \( 1 + 7.20T + 79T^{2} \)
83 \( 1 + 7.39T + 83T^{2} \)
89 \( 1 - 0.838T + 89T^{2} \)
97 \( 1 - 0.475T + 97T^{2} \)
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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

−7.63637052640872530801010778203, −6.47028897647393562458970710527, −6.13787390305214112790801999054, −5.33906718597048277065143609382, −4.49386020815810041381238353436, −4.30279468198870210611714088132, −2.64579118216129940511804190376, −2.39479981962883720170066297438, −1.26319367492338082619405631198, 0, 1.26319367492338082619405631198, 2.39479981962883720170066297438, 2.64579118216129940511804190376, 4.30279468198870210611714088132, 4.49386020815810041381238353436, 5.33906718597048277065143609382, 6.13787390305214112790801999054, 6.47028897647393562458970710527, 7.63637052640872530801010778203

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