Properties

Label 2-8048-1.1-c1-0-5
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.95·3-s + 0.296·5-s − 1.50·7-s + 5.72·9-s − 1.08·11-s − 3.03·13-s − 0.874·15-s − 1.39·17-s − 6.49·19-s + 4.43·21-s − 6.01·23-s − 4.91·25-s − 8.03·27-s + 2.16·29-s − 4.62·31-s + 3.19·33-s − 0.444·35-s + 3.14·37-s + 8.95·39-s − 3.26·41-s + 3.55·43-s + 1.69·45-s − 0.543·47-s − 4.74·49-s + 4.10·51-s − 9.68·53-s − 0.319·55-s + ⋯
L(s)  = 1  − 1.70·3-s + 0.132·5-s − 0.567·7-s + 1.90·9-s − 0.325·11-s − 0.841·13-s − 0.225·15-s − 0.337·17-s − 1.49·19-s + 0.966·21-s − 1.25·23-s − 0.982·25-s − 1.54·27-s + 0.401·29-s − 0.830·31-s + 0.555·33-s − 0.0750·35-s + 0.517·37-s + 1.43·39-s − 0.510·41-s + 0.542·43-s + 0.252·45-s − 0.0792·47-s − 0.678·49-s + 0.574·51-s − 1.33·53-s − 0.0431·55-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: \(0\)
Selberg data: \((2,\ 8048,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1006654030\)
\(L(\frac12)\) \(\approx\) \(0.1006654030\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.95T + 3T^{2} \)
5 \( 1 - 0.296T + 5T^{2} \)
7 \( 1 + 1.50T + 7T^{2} \)
11 \( 1 + 1.08T + 11T^{2} \)
13 \( 1 + 3.03T + 13T^{2} \)
17 \( 1 + 1.39T + 17T^{2} \)
19 \( 1 + 6.49T + 19T^{2} \)
23 \( 1 + 6.01T + 23T^{2} \)
29 \( 1 - 2.16T + 29T^{2} \)
31 \( 1 + 4.62T + 31T^{2} \)
37 \( 1 - 3.14T + 37T^{2} \)
41 \( 1 + 3.26T + 41T^{2} \)
43 \( 1 - 3.55T + 43T^{2} \)
47 \( 1 + 0.543T + 47T^{2} \)
53 \( 1 + 9.68T + 53T^{2} \)
59 \( 1 + 8.24T + 59T^{2} \)
61 \( 1 - 6.05T + 61T^{2} \)
67 \( 1 - 8.90T + 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 - 2.05T + 73T^{2} \)
79 \( 1 + 9.79T + 79T^{2} \)
83 \( 1 - 6.12T + 83T^{2} \)
89 \( 1 + 9.53T + 89T^{2} \)
97 \( 1 - 0.238T + 97T^{2} \)
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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

−7.63591330110459322157191798603, −6.92880780847911236233220472075, −6.19535322926220828874578436028, −5.98702801607906444104732594064, −5.07973922143623741724674098577, −4.48881588171793215448381734762, −3.78495702066751164749183392002, −2.49557371897256557317259844476, −1.64728289228886557235302459312, −0.16640334111048308022118797027, 0.16640334111048308022118797027, 1.64728289228886557235302459312, 2.49557371897256557317259844476, 3.78495702066751164749183392002, 4.48881588171793215448381734762, 5.07973922143623741724674098577, 5.98702801607906444104732594064, 6.19535322926220828874578436028, 6.92880780847911236233220472075, 7.63591330110459322157191798603

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