Properties

Label 2-8032-1.1-c1-0-75
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  + 2.71·3-s − 2.26·5-s − 1.04·7-s + 4.37·9-s − 4.92·11-s − 0.968·13-s − 6.14·15-s + 6.40·17-s + 6.87·19-s − 2.83·21-s + 5.06·23-s + 0.116·25-s + 3.72·27-s − 6.88·29-s − 5.16·31-s − 13.3·33-s + 2.36·35-s + 0.595·37-s − 2.63·39-s − 0.634·41-s − 2.58·43-s − 9.88·45-s − 3.05·47-s − 5.90·49-s + 17.3·51-s + 9.87·53-s + 11.1·55-s + ⋯
L(s)  = 1  + 1.56·3-s − 1.01·5-s − 0.394·7-s + 1.45·9-s − 1.48·11-s − 0.268·13-s − 1.58·15-s + 1.55·17-s + 1.57·19-s − 0.619·21-s + 1.05·23-s + 0.0232·25-s + 0.716·27-s − 1.27·29-s − 0.928·31-s − 2.32·33-s + 0.399·35-s + 0.0978·37-s − 0.421·39-s − 0.0990·41-s − 0.394·43-s − 1.47·45-s − 0.445·47-s − 0.843·49-s + 2.43·51-s + 1.35·53-s + 1.50·55-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

\(L(1)\) \(\approx\) \(2.605706108\)
\(L(\frac12)\) \(\approx\) \(2.605706108\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.71T + 3T^{2} \)
5 \( 1 + 2.26T + 5T^{2} \)
7 \( 1 + 1.04T + 7T^{2} \)
11 \( 1 + 4.92T + 11T^{2} \)
13 \( 1 + 0.968T + 13T^{2} \)
17 \( 1 - 6.40T + 17T^{2} \)
19 \( 1 - 6.87T + 19T^{2} \)
23 \( 1 - 5.06T + 23T^{2} \)
29 \( 1 + 6.88T + 29T^{2} \)
31 \( 1 + 5.16T + 31T^{2} \)
37 \( 1 - 0.595T + 37T^{2} \)
41 \( 1 + 0.634T + 41T^{2} \)
43 \( 1 + 2.58T + 43T^{2} \)
47 \( 1 + 3.05T + 47T^{2} \)
53 \( 1 - 9.87T + 53T^{2} \)
59 \( 1 - 7.03T + 59T^{2} \)
61 \( 1 - 8.27T + 61T^{2} \)
67 \( 1 - 15.2T + 67T^{2} \)
71 \( 1 + 3.09T + 71T^{2} \)
73 \( 1 - 9.03T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 - 16.7T + 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + 12.4T + 97T^{2} \)
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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

−7.77850510252601438789574418328, −7.54732493016414036542969223125, −6.86696442143675472221373423620, −5.34780093158749557795356289326, −5.22110515617538630996218088635, −3.79156315587203384231887187987, −3.47895664640765515134128794945, −2.89728980073511560189463670689, −2.04084292669679291491075726301, −0.72229407202409208191408449331, 0.72229407202409208191408449331, 2.04084292669679291491075726301, 2.89728980073511560189463670689, 3.47895664640765515134128794945, 3.79156315587203384231887187987, 5.22110515617538630996218088635, 5.34780093158749557795356289326, 6.86696442143675472221373423620, 7.54732493016414036542969223125, 7.77850510252601438789574418328

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