Properties

Label 2-8048-1.1-c1-0-124
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  + 2.49·3-s + 1.49·5-s + 1.21·7-s + 3.23·9-s − 2.74·11-s + 0.565·13-s + 3.73·15-s − 4.52·17-s + 3.08·19-s + 3.04·21-s + 7.09·23-s − 2.76·25-s + 0.590·27-s + 8.11·29-s − 6.98·31-s − 6.85·33-s + 1.82·35-s + 9.39·37-s + 1.41·39-s + 4.23·41-s + 9.67·43-s + 4.83·45-s + 1.27·47-s − 5.51·49-s − 11.2·51-s − 2.47·53-s − 4.10·55-s + ⋯
L(s)  = 1  + 1.44·3-s + 0.668·5-s + 0.460·7-s + 1.07·9-s − 0.827·11-s + 0.156·13-s + 0.963·15-s − 1.09·17-s + 0.708·19-s + 0.664·21-s + 1.48·23-s − 0.553·25-s + 0.113·27-s + 1.50·29-s − 1.25·31-s − 1.19·33-s + 0.307·35-s + 1.54·37-s + 0.226·39-s + 0.661·41-s + 1.47·43-s + 0.720·45-s + 0.186·47-s − 0.787·49-s − 1.58·51-s − 0.340·53-s − 0.552·55-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

\(L(1)\) \(\approx\) \(4.360371001\)
\(L(\frac12)\) \(\approx\) \(4.360371001\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.49T + 3T^{2} \)
5 \( 1 - 1.49T + 5T^{2} \)
7 \( 1 - 1.21T + 7T^{2} \)
11 \( 1 + 2.74T + 11T^{2} \)
13 \( 1 - 0.565T + 13T^{2} \)
17 \( 1 + 4.52T + 17T^{2} \)
19 \( 1 - 3.08T + 19T^{2} \)
23 \( 1 - 7.09T + 23T^{2} \)
29 \( 1 - 8.11T + 29T^{2} \)
31 \( 1 + 6.98T + 31T^{2} \)
37 \( 1 - 9.39T + 37T^{2} \)
41 \( 1 - 4.23T + 41T^{2} \)
43 \( 1 - 9.67T + 43T^{2} \)
47 \( 1 - 1.27T + 47T^{2} \)
53 \( 1 + 2.47T + 53T^{2} \)
59 \( 1 - 1.42T + 59T^{2} \)
61 \( 1 - 4.26T + 61T^{2} \)
67 \( 1 - 10.7T + 67T^{2} \)
71 \( 1 - 9.52T + 71T^{2} \)
73 \( 1 + 4.26T + 73T^{2} \)
79 \( 1 + 7.04T + 79T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 - 0.995T + 89T^{2} \)
97 \( 1 + 3.36T + 97T^{2} \)
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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

−7.86758036239150489992752884645, −7.39796034788867138682239197555, −6.56252080060820349447538992772, −5.70116741013257422184657567431, −4.95028057065455366032983615618, −4.23085507020237415050581415701, −3.29332013427959843882807231567, −2.55147282314096214332323145209, −2.12010330120506518859145390191, −0.986225005854791210151237000340, 0.986225005854791210151237000340, 2.12010330120506518859145390191, 2.55147282314096214332323145209, 3.29332013427959843882807231567, 4.23085507020237415050581415701, 4.95028057065455366032983615618, 5.70116741013257422184657567431, 6.56252080060820349447538992772, 7.39796034788867138682239197555, 7.86758036239150489992752884645

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