Properties

Label 2-8032-1.1-c1-0-62
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  + 1.18·3-s − 0.0964·5-s − 1.78·7-s − 1.59·9-s + 1.62·11-s − 0.858·13-s − 0.114·15-s − 0.953·17-s + 3.91·19-s − 2.11·21-s − 1.65·23-s − 4.99·25-s − 5.44·27-s + 8.35·29-s − 6.03·31-s + 1.92·33-s + 0.171·35-s + 10.3·37-s − 1.01·39-s − 5.38·41-s − 1.55·43-s + 0.153·45-s + 0.729·47-s − 3.81·49-s − 1.12·51-s + 2.02·53-s − 0.156·55-s + ⋯
L(s)  = 1  + 0.683·3-s − 0.0431·5-s − 0.674·7-s − 0.532·9-s + 0.489·11-s − 0.238·13-s − 0.0294·15-s − 0.231·17-s + 0.898·19-s − 0.461·21-s − 0.344·23-s − 0.998·25-s − 1.04·27-s + 1.55·29-s − 1.08·31-s + 0.334·33-s + 0.0290·35-s + 1.70·37-s − 0.162·39-s − 0.840·41-s − 0.237·43-s + 0.0229·45-s + 0.106·47-s − 0.545·49-s − 0.158·51-s + 0.278·53-s − 0.0211·55-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

\(L(1)\) \(\approx\) \(2.000705023\)
\(L(\frac12)\) \(\approx\) \(2.000705023\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.18T + 3T^{2} \)
5 \( 1 + 0.0964T + 5T^{2} \)
7 \( 1 + 1.78T + 7T^{2} \)
11 \( 1 - 1.62T + 11T^{2} \)
13 \( 1 + 0.858T + 13T^{2} \)
17 \( 1 + 0.953T + 17T^{2} \)
19 \( 1 - 3.91T + 19T^{2} \)
23 \( 1 + 1.65T + 23T^{2} \)
29 \( 1 - 8.35T + 29T^{2} \)
31 \( 1 + 6.03T + 31T^{2} \)
37 \( 1 - 10.3T + 37T^{2} \)
41 \( 1 + 5.38T + 41T^{2} \)
43 \( 1 + 1.55T + 43T^{2} \)
47 \( 1 - 0.729T + 47T^{2} \)
53 \( 1 - 2.02T + 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 - 6.10T + 61T^{2} \)
67 \( 1 + 3.88T + 67T^{2} \)
71 \( 1 - 5.75T + 71T^{2} \)
73 \( 1 - 12.8T + 73T^{2} \)
79 \( 1 - 3.04T + 79T^{2} \)
83 \( 1 + 0.674T + 83T^{2} \)
89 \( 1 + 6.71T + 89T^{2} \)
97 \( 1 - 5.60T + 97T^{2} \)
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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

−7.983528731546025726296700005214, −7.16154777175338343534941274854, −6.49377259639792312687106710912, −5.81172510429059115577539809871, −5.07132785120936502818853925478, −4.05283145743056227555781226062, −3.46515207647898826652309155139, −2.73670164635306109900267266046, −1.96697435916831042672006555201, −0.65578011700716576167069058225, 0.65578011700716576167069058225, 1.96697435916831042672006555201, 2.73670164635306109900267266046, 3.46515207647898826652309155139, 4.05283145743056227555781226062, 5.07132785120936502818853925478, 5.81172510429059115577539809871, 6.49377259639792312687106710912, 7.16154777175338343534941274854, 7.983528731546025726296700005214

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