Properties

Label 2-8048-1.1-c1-0-205
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s − 2·9-s + 5·11-s + 3·13-s − 6·19-s − 21-s − 23-s − 5·25-s − 5·27-s − 2·29-s − 4·31-s + 5·33-s + 8·37-s + 3·39-s − 4·41-s + 5·43-s + 47-s − 6·49-s − 6·57-s − 12·59-s − 61-s + 2·63-s + 9·67-s − 69-s − 6·71-s − 10·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s − 2/3·9-s + 1.50·11-s + 0.832·13-s − 1.37·19-s − 0.218·21-s − 0.208·23-s − 25-s − 0.962·27-s − 0.371·29-s − 0.718·31-s + 0.870·33-s + 1.31·37-s + 0.480·39-s − 0.624·41-s + 0.762·43-s + 0.145·47-s − 6/7·49-s − 0.794·57-s − 1.56·59-s − 0.128·61-s + 0.251·63-s + 1.09·67-s − 0.120·69-s − 0.712·71-s − 1.17·73-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: \(1\)
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 \( 1 - T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 17 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.61775716028495856287934743747, −6.63295430984178026366780363106, −6.18660130115353315343455553749, −5.62632193761852197215627061033, −4.34123130773115903853729330202, −3.88166107379104389380005593762, −3.19693918256683649792833646185, −2.22566158849502907791506143663, −1.41508612883361234288768071756, 0, 1.41508612883361234288768071756, 2.22566158849502907791506143663, 3.19693918256683649792833646185, 3.88166107379104389380005593762, 4.34123130773115903853729330202, 5.62632193761852197215627061033, 6.18660130115353315343455553749, 6.63295430984178026366780363106, 7.61775716028495856287934743747

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