Properties

Label 2-8032-1.1-c1-0-94
Degree $2$
Conductor $8032$
Sign $1$
Analytic cond. $64.1358$
Root an. cond. $8.00848$
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  − 2.60·3-s + 0.822·5-s + 3.53·7-s + 3.79·9-s + 2.95·11-s + 3.38·13-s − 2.14·15-s − 1.72·17-s − 0.518·19-s − 9.21·21-s + 5.52·23-s − 4.32·25-s − 2.06·27-s − 0.947·29-s + 1.67·31-s − 7.69·33-s + 2.90·35-s + 6.48·37-s − 8.83·39-s + 3.34·41-s + 5.45·43-s + 3.11·45-s − 10.5·47-s + 5.49·49-s + 4.49·51-s + 12.3·53-s + 2.42·55-s + ⋯
L(s)  = 1  − 1.50·3-s + 0.367·5-s + 1.33·7-s + 1.26·9-s + 0.890·11-s + 0.939·13-s − 0.553·15-s − 0.418·17-s − 0.119·19-s − 2.01·21-s + 1.15·23-s − 0.864·25-s − 0.396·27-s − 0.175·29-s + 0.301·31-s − 1.33·33-s + 0.491·35-s + 1.06·37-s − 1.41·39-s + 0.523·41-s + 0.832·43-s + 0.465·45-s − 1.53·47-s + 0.785·49-s + 0.629·51-s + 1.69·53-s + 0.327·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8032\)    =    \(2^{5} \cdot 251\)
Sign: $1$
Analytic conductor: \(64.1358\)
Root analytic conductor: \(8.00848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8032,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.873859681\)
\(L(\frac12)\) \(\approx\) \(1.873859681\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
251 \( 1 + T \)
good3 \( 1 + 2.60T + 3T^{2} \)
5 \( 1 - 0.822T + 5T^{2} \)
7 \( 1 - 3.53T + 7T^{2} \)
11 \( 1 - 2.95T + 11T^{2} \)
13 \( 1 - 3.38T + 13T^{2} \)
17 \( 1 + 1.72T + 17T^{2} \)
19 \( 1 + 0.518T + 19T^{2} \)
23 \( 1 - 5.52T + 23T^{2} \)
29 \( 1 + 0.947T + 29T^{2} \)
31 \( 1 - 1.67T + 31T^{2} \)
37 \( 1 - 6.48T + 37T^{2} \)
41 \( 1 - 3.34T + 41T^{2} \)
43 \( 1 - 5.45T + 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 - 12.3T + 53T^{2} \)
59 \( 1 - 13.0T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 - 0.944T + 67T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 + 14.3T + 73T^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 - 5.73T + 83T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + 7.94T + 97T^{2} \)
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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

−7.81800194846636819271359174783, −6.75072292075402836413172909246, −6.54594072961005380360499182134, −5.50110977702056980377823971961, −5.36471541899600610355269145613, −4.34953225952344158374578867573, −3.92060792958359218252084939356, −2.43716253358201088359709048174, −1.42448891228927796409036055804, −0.839427798772136795108550308667, 0.839427798772136795108550308667, 1.42448891228927796409036055804, 2.43716253358201088359709048174, 3.92060792958359218252084939356, 4.34953225952344158374578867573, 5.36471541899600610355269145613, 5.50110977702056980377823971961, 6.54594072961005380360499182134, 6.75072292075402836413172909246, 7.81800194846636819271359174783

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