Properties

Label 2-4024-1.1-c1-0-89
Degree $2$
Conductor $4024$
Sign $-1$
Analytic cond. $32.1318$
Root an. cond. $5.66849$
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  + 0.454·3-s + 0.622·5-s − 4.14·7-s − 2.79·9-s + 1.04·11-s + 4.67·13-s + 0.282·15-s + 1.91·17-s + 6.57·19-s − 1.88·21-s − 2.26·23-s − 4.61·25-s − 2.63·27-s − 9.03·29-s − 1.46·31-s + 0.476·33-s − 2.58·35-s + 6.96·37-s + 2.12·39-s − 0.613·41-s − 3.34·43-s − 1.73·45-s − 2.36·47-s + 10.1·49-s + 0.872·51-s + 0.528·53-s + 0.652·55-s + ⋯
L(s)  = 1  + 0.262·3-s + 0.278·5-s − 1.56·7-s − 0.931·9-s + 0.315·11-s + 1.29·13-s + 0.0730·15-s + 0.465·17-s + 1.50·19-s − 0.411·21-s − 0.472·23-s − 0.922·25-s − 0.506·27-s − 1.67·29-s − 0.263·31-s + 0.0828·33-s − 0.436·35-s + 1.14·37-s + 0.339·39-s − 0.0957·41-s − 0.509·43-s − 0.259·45-s − 0.345·47-s + 1.45·49-s + 0.122·51-s + 0.0725·53-s + 0.0879·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $-1$
Analytic conductor: \(32.1318\)
Root analytic conductor: \(5.66849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4024,\ (\ :1/2),\ -1)\)

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 \( 1 \)
503 \( 1 + T \)
good3 \( 1 - 0.454T + 3T^{2} \)
5 \( 1 - 0.622T + 5T^{2} \)
7 \( 1 + 4.14T + 7T^{2} \)
11 \( 1 - 1.04T + 11T^{2} \)
13 \( 1 - 4.67T + 13T^{2} \)
17 \( 1 - 1.91T + 17T^{2} \)
19 \( 1 - 6.57T + 19T^{2} \)
23 \( 1 + 2.26T + 23T^{2} \)
29 \( 1 + 9.03T + 29T^{2} \)
31 \( 1 + 1.46T + 31T^{2} \)
37 \( 1 - 6.96T + 37T^{2} \)
41 \( 1 + 0.613T + 41T^{2} \)
43 \( 1 + 3.34T + 43T^{2} \)
47 \( 1 + 2.36T + 47T^{2} \)
53 \( 1 - 0.528T + 53T^{2} \)
59 \( 1 + 11.9T + 59T^{2} \)
61 \( 1 - 4.74T + 61T^{2} \)
67 \( 1 + 16.2T + 67T^{2} \)
71 \( 1 - 8.46T + 71T^{2} \)
73 \( 1 - 6.36T + 73T^{2} \)
79 \( 1 + 2.85T + 79T^{2} \)
83 \( 1 + 1.00T + 83T^{2} \)
89 \( 1 + 13.9T + 89T^{2} \)
97 \( 1 - 5.06T + 97T^{2} \)
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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

−8.050682132281943034085863609684, −7.42308075801039732079748820863, −6.38737664104409529197407064983, −5.94569001613030223522468615089, −5.38891811251504682314301968239, −3.85228510492910644323313460030, −3.45791425026317975758784840355, −2.68786055385232116068846834561, −1.40923396179149905627602337625, 0, 1.40923396179149905627602337625, 2.68786055385232116068846834561, 3.45791425026317975758784840355, 3.85228510492910644323313460030, 5.38891811251504682314301968239, 5.94569001613030223522468615089, 6.38737664104409529197407064983, 7.42308075801039732079748820863, 8.050682132281943034085863609684

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