Properties

Label 2-8048-1.1-c1-0-208
Degree $2$
Conductor $8048$
Sign $1$
Analytic cond. $64.2636$
Root an. cond. $8.01645$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 2·5-s − 3·7-s + 6·9-s − 3·11-s + 5·13-s + 6·15-s − 8·17-s − 4·19-s + 9·21-s + 5·23-s − 25-s − 9·27-s + 2·31-s + 9·33-s + 6·35-s + 4·37-s − 15·39-s − 10·41-s + 43-s − 12·45-s + 3·47-s + 2·49-s + 24·51-s − 12·53-s + 6·55-s + 12·57-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.894·5-s − 1.13·7-s + 2·9-s − 0.904·11-s + 1.38·13-s + 1.54·15-s − 1.94·17-s − 0.917·19-s + 1.96·21-s + 1.04·23-s − 1/5·25-s − 1.73·27-s + 0.359·31-s + 1.56·33-s + 1.01·35-s + 0.657·37-s − 2.40·39-s − 1.56·41-s + 0.152·43-s − 1.78·45-s + 0.437·47-s + 2/7·49-s + 3.36·51-s − 1.64·53-s + 0.809·55-s + 1.58·57-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 8048,\ (\ :1/2),\ 1)\)

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 \( 1 + p T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−6.86056801424133015678012927175, −6.30648419214759360674079453609, −6.05765324497018397512711719092, −4.94866708471461602724793461904, −4.46434431345024855828052664238, −3.72014654214392512365142564104, −2.78509413701889502155122554479, −1.42493498312979270636249615972, 0, 0, 1.42493498312979270636249615972, 2.78509413701889502155122554479, 3.72014654214392512365142564104, 4.46434431345024855828052664238, 4.94866708471461602724793461904, 6.05765324497018397512711719092, 6.30648419214759360674079453609, 6.86056801424133015678012927175

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