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Properties

Label 2-32192-1.1-c1-0-25
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 7-s − 2·9-s − 5·11-s + 5·13-s − 6·19-s + 21-s − 23-s − 5·25-s + 5·27-s − 6·29-s − 8·31-s + 5·33-s + 4·37-s − 5·39-s − 5·43-s + 9·47-s − 6·49-s − 4·53-s + 6·57-s − 12·59-s − 7·61-s + 2·63-s − 9·67-s + 69-s + 2·71-s + 14·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s − 2/3·9-s − 1.50·11-s + 1.38·13-s − 1.37·19-s + 0.218·21-s − 0.208·23-s − 25-s + 0.962·27-s − 1.11·29-s − 1.43·31-s + 0.870·33-s + 0.657·37-s − 0.800·39-s − 0.762·43-s + 1.31·47-s − 6/7·49-s − 0.549·53-s + 0.794·57-s − 1.56·59-s − 0.896·61-s + 0.251·63-s − 1.09·67-s + 0.120·69-s + 0.237·71-s + 1.63·73-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: \(2\)
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 \( 1 + T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−15.53070805089921, −15.22517950342267, −14.58301266691583, −13.83856768625692, −13.39590764674736, −12.97109303053845, −12.46914733982101, −11.87819714346249, −11.01070798461567, −10.90441877167034, −10.56814001386507, −9.619012761577817, −9.183244300519249, −8.437173460270366, −8.029907809799748, −7.483567884078817, −6.586539351959499, −6.096454013320363, −5.669860149663668, −5.172764836887240, −4.291172219416423, −3.685315940475521, −2.977334391288798, −2.246124644277652, −1.459819293519669, 0, 0, 1.459819293519669, 2.246124644277652, 2.977334391288798, 3.685315940475521, 4.291172219416423, 5.172764836887240, 5.669860149663668, 6.096454013320363, 6.586539351959499, 7.483567884078817, 8.029907809799748, 8.437173460270366, 9.183244300519249, 9.619012761577817, 10.56814001386507, 10.90441877167034, 11.01070798461567, 11.87819714346249, 12.46914733982101, 12.97109303053845, 13.39590764674736, 13.83856768625692, 14.58301266691583, 15.22517950342267, 15.53070805089921

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