Properties

Label 2-8048-1.1-c1-0-168
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  − 0.659·3-s + 0.997·5-s − 0.366·7-s − 2.56·9-s − 2.34·11-s + 1.58·13-s − 0.657·15-s + 0.164·17-s + 2.94·19-s + 0.241·21-s + 6.49·23-s − 4.00·25-s + 3.66·27-s − 7.89·29-s + 7.02·31-s + 1.54·33-s − 0.365·35-s − 5.39·37-s − 1.04·39-s − 4.46·41-s − 8.37·43-s − 2.56·45-s + 12.8·47-s − 6.86·49-s − 0.108·51-s + 12.7·53-s − 2.33·55-s + ⋯
L(s)  = 1  − 0.380·3-s + 0.446·5-s − 0.138·7-s − 0.855·9-s − 0.706·11-s + 0.439·13-s − 0.169·15-s + 0.0398·17-s + 0.674·19-s + 0.0527·21-s + 1.35·23-s − 0.800·25-s + 0.705·27-s − 1.46·29-s + 1.26·31-s + 0.268·33-s − 0.0618·35-s − 0.886·37-s − 0.167·39-s − 0.697·41-s − 1.27·43-s − 0.381·45-s + 1.87·47-s − 0.980·49-s − 0.0151·51-s + 1.74·53-s − 0.315·55-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 \( 1 + 0.659T + 3T^{2} \)
5 \( 1 - 0.997T + 5T^{2} \)
7 \( 1 + 0.366T + 7T^{2} \)
11 \( 1 + 2.34T + 11T^{2} \)
13 \( 1 - 1.58T + 13T^{2} \)
17 \( 1 - 0.164T + 17T^{2} \)
19 \( 1 - 2.94T + 19T^{2} \)
23 \( 1 - 6.49T + 23T^{2} \)
29 \( 1 + 7.89T + 29T^{2} \)
31 \( 1 - 7.02T + 31T^{2} \)
37 \( 1 + 5.39T + 37T^{2} \)
41 \( 1 + 4.46T + 41T^{2} \)
43 \( 1 + 8.37T + 43T^{2} \)
47 \( 1 - 12.8T + 47T^{2} \)
53 \( 1 - 12.7T + 53T^{2} \)
59 \( 1 + 9.64T + 59T^{2} \)
61 \( 1 - 9.98T + 61T^{2} \)
67 \( 1 - 6.58T + 67T^{2} \)
71 \( 1 + 9.53T + 71T^{2} \)
73 \( 1 - 2.32T + 73T^{2} \)
79 \( 1 + 2.51T + 79T^{2} \)
83 \( 1 - 2.90T + 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 - 0.261T + 97T^{2} \)
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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

−7.40749932735260583157468538750, −6.76690546424513837130031049813, −5.95732013391852338032927194782, −5.42426461411850992494333162964, −4.96491251758354275295752160598, −3.80012353095215595221368010430, −3.06127350774853901842900315547, −2.30090810182074356840844086618, −1.18489289159193189254183927522, 0, 1.18489289159193189254183927522, 2.30090810182074356840844086618, 3.06127350774853901842900315547, 3.80012353095215595221368010430, 4.96491251758354275295752160598, 5.42426461411850992494333162964, 5.95732013391852338032927194782, 6.76690546424513837130031049813, 7.40749932735260583157468538750

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