Properties

Label 2-8048-1.1-c1-0-156
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  + 1.85·3-s + 2.88·5-s − 1.11·7-s + 0.447·9-s + 4.87·11-s + 3.77·13-s + 5.36·15-s − 0.0587·17-s + 2.60·19-s − 2.06·21-s + 2.33·23-s + 3.34·25-s − 4.73·27-s + 6.17·29-s + 4.36·31-s + 9.05·33-s − 3.21·35-s − 4.47·37-s + 7.01·39-s − 2.97·41-s + 10.9·43-s + 1.29·45-s − 8.04·47-s − 5.76·49-s − 0.109·51-s − 11.7·53-s + 14.0·55-s + ⋯
L(s)  = 1  + 1.07·3-s + 1.29·5-s − 0.420·7-s + 0.149·9-s + 1.47·11-s + 1.04·13-s + 1.38·15-s − 0.0142·17-s + 0.596·19-s − 0.450·21-s + 0.486·23-s + 0.669·25-s − 0.912·27-s + 1.14·29-s + 0.784·31-s + 1.57·33-s − 0.543·35-s − 0.735·37-s + 1.12·39-s − 0.464·41-s + 1.66·43-s + 0.192·45-s − 1.17·47-s − 0.823·49-s − 0.0152·51-s − 1.60·53-s + 1.90·55-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

\(L(1)\) \(\approx\) \(4.637420301\)
\(L(\frac12)\) \(\approx\) \(4.637420301\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.85T + 3T^{2} \)
5 \( 1 - 2.88T + 5T^{2} \)
7 \( 1 + 1.11T + 7T^{2} \)
11 \( 1 - 4.87T + 11T^{2} \)
13 \( 1 - 3.77T + 13T^{2} \)
17 \( 1 + 0.0587T + 17T^{2} \)
19 \( 1 - 2.60T + 19T^{2} \)
23 \( 1 - 2.33T + 23T^{2} \)
29 \( 1 - 6.17T + 29T^{2} \)
31 \( 1 - 4.36T + 31T^{2} \)
37 \( 1 + 4.47T + 37T^{2} \)
41 \( 1 + 2.97T + 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 + 8.04T + 47T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 - 1.31T + 59T^{2} \)
61 \( 1 - 7.26T + 61T^{2} \)
67 \( 1 - 5.79T + 67T^{2} \)
71 \( 1 - 0.407T + 71T^{2} \)
73 \( 1 - 1.44T + 73T^{2} \)
79 \( 1 + 9.65T + 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 + 4.02T + 97T^{2} \)
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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

−8.097299546735878133638598847107, −6.97601755023333886579371291836, −6.43203014066648237814202867887, −5.95863284770633896350026147492, −5.07456982892870620787526250365, −4.07095903865295253289773388920, −3.33940330495128956277511723269, −2.76129545910728488365386264492, −1.77777378314501831459649112197, −1.11915244386668743520572601330, 1.11915244386668743520572601330, 1.77777378314501831459649112197, 2.76129545910728488365386264492, 3.33940330495128956277511723269, 4.07095903865295253289773388920, 5.07456982892870620787526250365, 5.95863284770633896350026147492, 6.43203014066648237814202867887, 6.97601755023333886579371291836, 8.097299546735878133638598847107

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