Properties

Label 2-8027-1.1-c1-0-255
Degree $2$
Conductor $8027$
Sign $1$
Analytic cond. $64.0959$
Root an. cond. $8.00599$
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  − 2.62·2-s + 1.97·3-s + 4.89·4-s − 2.67·5-s − 5.17·6-s + 3.82·7-s − 7.61·8-s + 0.885·9-s + 7.01·10-s + 5.21·11-s + 9.65·12-s − 1.19·13-s − 10.0·14-s − 5.26·15-s + 10.2·16-s + 3.34·17-s − 2.32·18-s + 0.211·19-s − 13.0·20-s + 7.53·21-s − 13.6·22-s + 23-s − 15.0·24-s + 2.13·25-s + 3.12·26-s − 4.16·27-s + 18.7·28-s + ⋯
L(s)  = 1  − 1.85·2-s + 1.13·3-s + 2.44·4-s − 1.19·5-s − 2.11·6-s + 1.44·7-s − 2.69·8-s + 0.295·9-s + 2.21·10-s + 1.57·11-s + 2.78·12-s − 0.330·13-s − 2.68·14-s − 1.35·15-s + 2.55·16-s + 0.811·17-s − 0.548·18-s + 0.0484·19-s − 2.92·20-s + 1.64·21-s − 2.92·22-s + 0.208·23-s − 3.06·24-s + 0.426·25-s + 0.613·26-s − 0.802·27-s + 3.54·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8027\)    =    \(23 \cdot 349\)
Sign: $1$
Analytic conductor: \(64.0959\)
Root analytic conductor: \(8.00599\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8027,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.405950814\)
\(L(\frac12)\) \(\approx\) \(1.405950814\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 - T \)
349 \( 1 + T \)
good2 \( 1 + 2.62T + 2T^{2} \)
3 \( 1 - 1.97T + 3T^{2} \)
5 \( 1 + 2.67T + 5T^{2} \)
7 \( 1 - 3.82T + 7T^{2} \)
11 \( 1 - 5.21T + 11T^{2} \)
13 \( 1 + 1.19T + 13T^{2} \)
17 \( 1 - 3.34T + 17T^{2} \)
19 \( 1 - 0.211T + 19T^{2} \)
29 \( 1 + 0.308T + 29T^{2} \)
31 \( 1 - 7.17T + 31T^{2} \)
37 \( 1 + 5.24T + 37T^{2} \)
41 \( 1 + 5.30T + 41T^{2} \)
43 \( 1 - 3.18T + 43T^{2} \)
47 \( 1 - 0.360T + 47T^{2} \)
53 \( 1 - 1.71T + 53T^{2} \)
59 \( 1 + 1.58T + 59T^{2} \)
61 \( 1 - 9.22T + 61T^{2} \)
67 \( 1 - 9.27T + 67T^{2} \)
71 \( 1 - 0.623T + 71T^{2} \)
73 \( 1 - 5.13T + 73T^{2} \)
79 \( 1 - 8.42T + 79T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 - 1.12T + 89T^{2} \)
97 \( 1 - 6.84T + 97T^{2} \)
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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

−8.231624896174031316450096771434, −7.41736515946040519450178865240, −7.11157627639716332640918255787, −6.13973016373741136376832556016, −4.94798385993200618889164080381, −3.94435329624201680868303370576, −3.32977704803609389281056054691, −2.32871656772976528882185672849, −1.56823479470033699606219585889, −0.78442537708931156802256888119, 0.78442537708931156802256888119, 1.56823479470033699606219585889, 2.32871656772976528882185672849, 3.32977704803609389281056054691, 3.94435329624201680868303370576, 4.94798385993200618889164080381, 6.13973016373741136376832556016, 7.11157627639716332640918255787, 7.41736515946040519450178865240, 8.231624896174031316450096771434

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