Properties

Label 2-8032-1.1-c1-0-224
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.74·3-s − 4.14·5-s + 3.66·7-s + 4.51·9-s − 2.70·11-s + 4.94·13-s − 11.3·15-s − 1.67·17-s − 6.88·19-s + 10.0·21-s − 4.06·23-s + 12.1·25-s + 4.15·27-s − 6.03·29-s + 0.559·31-s − 7.41·33-s − 15.1·35-s − 1.38·37-s + 13.5·39-s − 7.23·41-s − 2.36·43-s − 18.7·45-s − 9.14·47-s + 6.41·49-s − 4.59·51-s + 11.6·53-s + 11.2·55-s + ⋯
L(s)  = 1  + 1.58·3-s − 1.85·5-s + 1.38·7-s + 1.50·9-s − 0.815·11-s + 1.37·13-s − 2.93·15-s − 0.406·17-s − 1.57·19-s + 2.19·21-s − 0.848·23-s + 2.43·25-s + 0.798·27-s − 1.12·29-s + 0.100·31-s − 1.29·33-s − 2.56·35-s − 0.227·37-s + 2.17·39-s − 1.13·41-s − 0.360·43-s − 2.79·45-s − 1.33·47-s + 0.915·49-s − 0.643·51-s + 1.60·53-s + 1.51·55-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: \(1\)
Selberg data: \((2,\ 8032,\ (\ :1/2),\ -1)\)

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 \( 1 \)
251 \( 1 - T \)
good3 \( 1 - 2.74T + 3T^{2} \)
5 \( 1 + 4.14T + 5T^{2} \)
7 \( 1 - 3.66T + 7T^{2} \)
11 \( 1 + 2.70T + 11T^{2} \)
13 \( 1 - 4.94T + 13T^{2} \)
17 \( 1 + 1.67T + 17T^{2} \)
19 \( 1 + 6.88T + 19T^{2} \)
23 \( 1 + 4.06T + 23T^{2} \)
29 \( 1 + 6.03T + 29T^{2} \)
31 \( 1 - 0.559T + 31T^{2} \)
37 \( 1 + 1.38T + 37T^{2} \)
41 \( 1 + 7.23T + 41T^{2} \)
43 \( 1 + 2.36T + 43T^{2} \)
47 \( 1 + 9.14T + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + 7.42T + 59T^{2} \)
61 \( 1 - 1.42T + 61T^{2} \)
67 \( 1 + 7.90T + 67T^{2} \)
71 \( 1 + 1.27T + 71T^{2} \)
73 \( 1 + 12.5T + 73T^{2} \)
79 \( 1 - 12.5T + 79T^{2} \)
83 \( 1 - 5.14T + 83T^{2} \)
89 \( 1 - 8.12T + 89T^{2} \)
97 \( 1 + 6.29T + 97T^{2} \)
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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

−7.80471757885244248203002191035, −7.23624352482979776598742836351, −6.28468534942895165977165538902, −5.04486959573059561468641102588, −4.34618891690397263372622364704, −3.84983985460575492898721071150, −3.29296336360466778261979072311, −2.25853785723344043573027353776, −1.53213774656440917743530021915, 0, 1.53213774656440917743530021915, 2.25853785723344043573027353776, 3.29296336360466778261979072311, 3.84983985460575492898721071150, 4.34618891690397263372622364704, 5.04486959573059561468641102588, 6.28468534942895165977165538902, 7.23624352482979776598742836351, 7.80471757885244248203002191035

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