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Properties

Label 2-32192-1.1-c1-0-9
Degree $2$
Conductor $32192$
Sign $-1$
Analytic cond. $257.054$
Root an. cond. $16.0329$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 2·5-s − 3·7-s + 6·9-s − 3·11-s − 3·13-s − 6·15-s + 2·17-s + 6·19-s + 9·21-s − 3·23-s − 25-s − 9·27-s − 2·29-s − 6·31-s + 9·33-s − 6·35-s + 2·37-s + 9·39-s + 10·41-s − 3·43-s + 12·45-s − 9·47-s + 2·49-s − 6·51-s − 2·53-s − 6·55-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.894·5-s − 1.13·7-s + 2·9-s − 0.904·11-s − 0.832·13-s − 1.54·15-s + 0.485·17-s + 1.37·19-s + 1.96·21-s − 0.625·23-s − 1/5·25-s − 1.73·27-s − 0.371·29-s − 1.07·31-s + 1.56·33-s − 1.01·35-s + 0.328·37-s + 1.44·39-s + 1.56·41-s − 0.457·43-s + 1.78·45-s − 1.31·47-s + 2/7·49-s − 0.840·51-s − 0.274·53-s − 0.809·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32192\)    =    \(2^{6} \cdot 503\)
Sign: $-1$
Analytic conductor: \(257.054\)
Root analytic conductor: \(16.0329\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 32192,\ (\ :1/2),\ -1)\)

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 \( 1 \)
503 \( 1 - T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−15.64928672004698, −14.74494430851638, −14.28637987004912, −13.40156574099274, −13.12766967368804, −12.67479142639501, −12.11465227887282, −11.67082533980098, −11.03952879534827, −10.47596965241074, −9.927560949580534, −9.661340343005779, −9.288677081529446, −8.009734961673217, −7.509949099681145, −6.916442538908587, −6.346298799576196, −5.814000347079174, −5.341466915707930, −5.090270439732395, −4.114686153445599, −3.343167450230864, −2.531705551938604, −1.730465332308105, −0.7429870939862395, 0, 0.7429870939862395, 1.730465332308105, 2.531705551938604, 3.343167450230864, 4.114686153445599, 5.090270439732395, 5.341466915707930, 5.814000347079174, 6.346298799576196, 6.916442538908587, 7.509949099681145, 8.009734961673217, 9.288677081529446, 9.661340343005779, 9.927560949580534, 10.47596965241074, 11.03952879534827, 11.67082533980098, 12.11465227887282, 12.67479142639501, 13.12766967368804, 13.40156574099274, 14.28637987004912, 14.74494430851638, 15.64928672004698

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