Properties

Label 2-8036-1.1-c1-0-133
Degree $2$
Conductor $8036$
Sign $-1$
Analytic cond. $64.1677$
Root an. cond. $8.01047$
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  + 2.10·3-s + 2.93·5-s + 1.42·9-s − 5.80·11-s + 1.53·13-s + 6.17·15-s − 5.69·17-s − 3.30·19-s − 2.90·23-s + 3.61·25-s − 3.31·27-s − 1.53·29-s + 6.66·31-s − 12.2·33-s − 11.3·37-s + 3.22·39-s − 41-s − 2.99·43-s + 4.17·45-s − 8.97·47-s − 11.9·51-s + 2.87·53-s − 17.0·55-s − 6.95·57-s − 9.75·59-s − 6.56·61-s + 4.50·65-s + ⋯
L(s)  = 1  + 1.21·3-s + 1.31·5-s + 0.474·9-s − 1.75·11-s + 0.425·13-s + 1.59·15-s − 1.38·17-s − 0.758·19-s − 0.606·23-s + 0.723·25-s − 0.638·27-s − 0.285·29-s + 1.19·31-s − 2.12·33-s − 1.86·37-s + 0.516·39-s − 0.156·41-s − 0.457·43-s + 0.622·45-s − 1.30·47-s − 1.67·51-s + 0.394·53-s − 2.29·55-s − 0.920·57-s − 1.27·59-s − 0.840·61-s + 0.558·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8036\)    =    \(2^{2} \cdot 7^{2} \cdot 41\)
Sign: $-1$
Analytic conductor: \(64.1677\)
Root analytic conductor: \(8.01047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8036,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
41 \( 1 + T \)
good3 \( 1 - 2.10T + 3T^{2} \)
5 \( 1 - 2.93T + 5T^{2} \)
11 \( 1 + 5.80T + 11T^{2} \)
13 \( 1 - 1.53T + 13T^{2} \)
17 \( 1 + 5.69T + 17T^{2} \)
19 \( 1 + 3.30T + 19T^{2} \)
23 \( 1 + 2.90T + 23T^{2} \)
29 \( 1 + 1.53T + 29T^{2} \)
31 \( 1 - 6.66T + 31T^{2} \)
37 \( 1 + 11.3T + 37T^{2} \)
43 \( 1 + 2.99T + 43T^{2} \)
47 \( 1 + 8.97T + 47T^{2} \)
53 \( 1 - 2.87T + 53T^{2} \)
59 \( 1 + 9.75T + 59T^{2} \)
61 \( 1 + 6.56T + 61T^{2} \)
67 \( 1 - 8.35T + 67T^{2} \)
71 \( 1 - 12.2T + 71T^{2} \)
73 \( 1 + 2.02T + 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 - 8.68T + 83T^{2} \)
89 \( 1 - 4.80T + 89T^{2} \)
97 \( 1 - 4.61T + 97T^{2} \)
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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

−7.73780479564205887674475858239, −6.69683752712186698341043355516, −6.21435183014252786488034265239, −5.32746531758607224524323123553, −4.76437234440689217188570516601, −3.70461905914397456173794945638, −2.86972937560735761675932595686, −2.21481893078472316433784577315, −1.81419812120355120535487194878, 0, 1.81419812120355120535487194878, 2.21481893078472316433784577315, 2.86972937560735761675932595686, 3.70461905914397456173794945638, 4.76437234440689217188570516601, 5.32746531758607224524323123553, 6.21435183014252786488034265239, 6.69683752712186698341043355516, 7.73780479564205887674475858239

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