Properties

Label 2-8048-1.1-c1-0-86
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.78·3-s − 0.988·5-s − 3.73·7-s + 0.196·9-s − 5.76·11-s + 4.01·13-s + 1.76·15-s + 6.16·17-s − 4.14·19-s + 6.68·21-s + 1.16·23-s − 4.02·25-s + 5.01·27-s + 5.75·29-s − 1.16·31-s + 10.3·33-s + 3.69·35-s − 9.17·37-s − 7.18·39-s − 7.67·41-s + 3.71·43-s − 0.194·45-s + 5.48·47-s + 6.96·49-s − 11.0·51-s + 1.03·53-s + 5.69·55-s + ⋯
L(s)  = 1  − 1.03·3-s − 0.441·5-s − 1.41·7-s + 0.0656·9-s − 1.73·11-s + 1.11·13-s + 0.456·15-s + 1.49·17-s − 0.950·19-s + 1.45·21-s + 0.243·23-s − 0.804·25-s + 0.964·27-s + 1.06·29-s − 0.209·31-s + 1.79·33-s + 0.624·35-s − 1.50·37-s − 1.14·39-s − 1.19·41-s + 0.566·43-s − 0.0290·45-s + 0.800·47-s + 0.995·49-s − 1.54·51-s + 0.141·53-s + 0.767·55-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: \(1\)
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 \( 1 + 1.78T + 3T^{2} \)
5 \( 1 + 0.988T + 5T^{2} \)
7 \( 1 + 3.73T + 7T^{2} \)
11 \( 1 + 5.76T + 11T^{2} \)
13 \( 1 - 4.01T + 13T^{2} \)
17 \( 1 - 6.16T + 17T^{2} \)
19 \( 1 + 4.14T + 19T^{2} \)
23 \( 1 - 1.16T + 23T^{2} \)
29 \( 1 - 5.75T + 29T^{2} \)
31 \( 1 + 1.16T + 31T^{2} \)
37 \( 1 + 9.17T + 37T^{2} \)
41 \( 1 + 7.67T + 41T^{2} \)
43 \( 1 - 3.71T + 43T^{2} \)
47 \( 1 - 5.48T + 47T^{2} \)
53 \( 1 - 1.03T + 53T^{2} \)
59 \( 1 - 5.95T + 59T^{2} \)
61 \( 1 + 2.55T + 61T^{2} \)
67 \( 1 - 9.37T + 67T^{2} \)
71 \( 1 - 4.55T + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 - 9.39T + 79T^{2} \)
83 \( 1 + 3.37T + 83T^{2} \)
89 \( 1 + 0.772T + 89T^{2} \)
97 \( 1 - 16.9T + 97T^{2} \)
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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

−7.38464947508818958900985830112, −6.62371358786439324705033527267, −6.07017510044687443346269737475, −5.47980853732491313531339764630, −4.92723674119492229468845380480, −3.71247094133442550592382770806, −3.28874554208523382192364355836, −2.33974853884419675898876576035, −0.811853427214395451238250922859, 0, 0.811853427214395451238250922859, 2.33974853884419675898876576035, 3.28874554208523382192364355836, 3.71247094133442550592382770806, 4.92723674119492229468845380480, 5.47980853732491313531339764630, 6.07017510044687443346269737475, 6.62371358786439324705033527267, 7.38464947508818958900985830112

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