Properties

Label 2-8048-1.1-c1-0-225
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.76·3-s − 2.37·5-s + 2.36·7-s + 4.64·9-s − 1.30·11-s − 3.45·13-s − 6.57·15-s − 3.40·17-s − 3.93·19-s + 6.53·21-s − 0.710·23-s + 0.657·25-s + 4.56·27-s + 5.88·29-s − 3.60·31-s − 3.61·33-s − 5.62·35-s + 7.19·37-s − 9.55·39-s + 7.35·41-s − 3.43·43-s − 11.0·45-s + 0.666·47-s − 1.41·49-s − 9.41·51-s + 1.77·53-s + 3.11·55-s + ⋯
L(s)  = 1  + 1.59·3-s − 1.06·5-s + 0.893·7-s + 1.54·9-s − 0.394·11-s − 0.958·13-s − 1.69·15-s − 0.825·17-s − 0.901·19-s + 1.42·21-s − 0.148·23-s + 0.131·25-s + 0.878·27-s + 1.09·29-s − 0.647·31-s − 0.629·33-s − 0.950·35-s + 1.18·37-s − 1.53·39-s + 1.14·41-s − 0.523·43-s − 1.64·45-s + 0.0972·47-s − 0.202·49-s − 1.31·51-s + 0.243·53-s + 0.419·55-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 \( 1 - 2.76T + 3T^{2} \)
5 \( 1 + 2.37T + 5T^{2} \)
7 \( 1 - 2.36T + 7T^{2} \)
11 \( 1 + 1.30T + 11T^{2} \)
13 \( 1 + 3.45T + 13T^{2} \)
17 \( 1 + 3.40T + 17T^{2} \)
19 \( 1 + 3.93T + 19T^{2} \)
23 \( 1 + 0.710T + 23T^{2} \)
29 \( 1 - 5.88T + 29T^{2} \)
31 \( 1 + 3.60T + 31T^{2} \)
37 \( 1 - 7.19T + 37T^{2} \)
41 \( 1 - 7.35T + 41T^{2} \)
43 \( 1 + 3.43T + 43T^{2} \)
47 \( 1 - 0.666T + 47T^{2} \)
53 \( 1 - 1.77T + 53T^{2} \)
59 \( 1 + 4.65T + 59T^{2} \)
61 \( 1 - 1.08T + 61T^{2} \)
67 \( 1 + 4.94T + 67T^{2} \)
71 \( 1 + 9.87T + 71T^{2} \)
73 \( 1 + 5.89T + 73T^{2} \)
79 \( 1 + 5.05T + 79T^{2} \)
83 \( 1 - 0.501T + 83T^{2} \)
89 \( 1 + 14.0T + 89T^{2} \)
97 \( 1 + 0.210T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.63905984058363412063756589230, −7.24579781696682213802995637189, −6.27521780894991739560589435883, −5.08548232854992960596935143013, −4.27746343895723798451517604638, −4.09854970808164510097708431493, −2.89641350083970004794848066747, −2.46843210895920392019481007562, −1.54705744694947460437669292511, 0, 1.54705744694947460437669292511, 2.46843210895920392019481007562, 2.89641350083970004794848066747, 4.09854970808164510097708431493, 4.27746343895723798451517604638, 5.08548232854992960596935143013, 6.27521780894991739560589435883, 7.24579781696682213802995637189, 7.63905984058363412063756589230

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