Properties

Label 2-8048-1.1-c1-0-19
Degree $2$
Conductor $8048$
Sign $1$
Analytic cond. $64.2636$
Root an. cond. $8.01645$
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  + 0.224·3-s − 2.01·5-s − 1.98·7-s − 2.94·9-s − 5.16·11-s + 4.72·13-s − 0.451·15-s + 3.77·17-s − 1.20·19-s − 0.445·21-s + 1.63·23-s − 0.951·25-s − 1.33·27-s − 8.63·29-s − 6.56·31-s − 1.16·33-s + 3.98·35-s + 1.73·37-s + 1.06·39-s + 0.973·41-s + 6.75·43-s + 5.93·45-s − 11.9·47-s − 3.06·49-s + 0.848·51-s − 8.24·53-s + 10.3·55-s + ⋯
L(s)  = 1  + 0.129·3-s − 0.899·5-s − 0.749·7-s − 0.983·9-s − 1.55·11-s + 1.30·13-s − 0.116·15-s + 0.915·17-s − 0.276·19-s − 0.0971·21-s + 0.340·23-s − 0.190·25-s − 0.257·27-s − 1.60·29-s − 1.17·31-s − 0.201·33-s + 0.674·35-s + 0.284·37-s + 0.169·39-s + 0.151·41-s + 1.02·43-s + 0.884·45-s − 1.74·47-s − 0.438·49-s + 0.118·51-s − 1.13·53-s + 1.40·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8048\)    =    \(2^{4} \cdot 503\)
Sign: $1$
Analytic conductor: \(64.2636\)
Root analytic conductor: \(8.01645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8048,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5527569011\)
\(L(\frac12)\) \(\approx\) \(0.5527569011\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.224T + 3T^{2} \)
5 \( 1 + 2.01T + 5T^{2} \)
7 \( 1 + 1.98T + 7T^{2} \)
11 \( 1 + 5.16T + 11T^{2} \)
13 \( 1 - 4.72T + 13T^{2} \)
17 \( 1 - 3.77T + 17T^{2} \)
19 \( 1 + 1.20T + 19T^{2} \)
23 \( 1 - 1.63T + 23T^{2} \)
29 \( 1 + 8.63T + 29T^{2} \)
31 \( 1 + 6.56T + 31T^{2} \)
37 \( 1 - 1.73T + 37T^{2} \)
41 \( 1 - 0.973T + 41T^{2} \)
43 \( 1 - 6.75T + 43T^{2} \)
47 \( 1 + 11.9T + 47T^{2} \)
53 \( 1 + 8.24T + 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 - 1.56T + 61T^{2} \)
67 \( 1 - 8.49T + 67T^{2} \)
71 \( 1 - 0.0603T + 71T^{2} \)
73 \( 1 + 3.47T + 73T^{2} \)
79 \( 1 - 6.07T + 79T^{2} \)
83 \( 1 + 14.2T + 83T^{2} \)
89 \( 1 - 0.498T + 89T^{2} \)
97 \( 1 + 8.81T + 97T^{2} \)
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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

−7.900268392419262929704744259025, −7.37292731378093520428039924364, −6.33893285693372080954905354360, −5.73390008680140050117931135421, −5.19766907297794744117944294309, −4.08858255552345737098985169280, −3.33264891913536988192524566496, −3.02005031918973557736866573305, −1.81840274505535517351576605699, −0.34731497961269972225936524578, 0.34731497961269972225936524578, 1.81840274505535517351576605699, 3.02005031918973557736866573305, 3.33264891913536988192524566496, 4.08858255552345737098985169280, 5.19766907297794744117944294309, 5.73390008680140050117931135421, 6.33893285693372080954905354360, 7.37292731378093520428039924364, 7.900268392419262929704744259025

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