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Properties

Label 2-32192-1.1-c1-0-29
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 − 2·5-s + 7-s − 2·9-s − 3·11-s − 3·13-s − 2·15-s − 2·17-s + 21-s − 3·23-s − 25-s − 5·27-s − 6·29-s + 4·31-s − 3·33-s − 2·35-s − 8·37-s − 3·39-s − 12·41-s − 7·43-s + 4·45-s + 3·47-s − 6·49-s − 2·51-s − 12·53-s + 6·55-s − 8·59-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 0.377·7-s − 2/3·9-s − 0.904·11-s − 0.832·13-s − 0.516·15-s − 0.485·17-s + 0.218·21-s − 0.625·23-s − 1/5·25-s − 0.962·27-s − 1.11·29-s + 0.718·31-s − 0.522·33-s − 0.338·35-s − 1.31·37-s − 0.480·39-s − 1.87·41-s − 1.06·43-s + 0.596·45-s + 0.437·47-s − 6/7·49-s − 0.280·51-s − 1.64·53-s + 0.809·55-s − 1.04·59-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 \( 1 + 2 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.56759211126399, −15.12125235701731, −14.48251095551464, −14.14184889974190, −13.44171399494937, −13.09582757976226, −12.25571497731528, −11.89740575651878, −11.37051863851824, −10.89878404121363, −10.16492691651870, −9.716135320468840, −8.989480200795182, −8.344284539306917, −8.067135959598829, −7.594878303107813, −6.955356554145519, −6.263048458713872, −5.367429471848584, −5.007046403078981, −4.296233151212802, −3.494785446907448, −3.102428227942435, −2.233052750144472, −1.703586165165640, 0, 0, 1.703586165165640, 2.233052750144472, 3.102428227942435, 3.494785446907448, 4.296233151212802, 5.007046403078981, 5.367429471848584, 6.263048458713872, 6.955356554145519, 7.594878303107813, 8.067135959598829, 8.344284539306917, 8.989480200795182, 9.716135320468840, 10.16492691651870, 10.89878404121363, 11.37051863851824, 11.89740575651878, 12.25571497731528, 13.09582757976226, 13.44171399494937, 14.14184889974190, 14.48251095551464, 15.12125235701731, 15.56759211126399

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