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Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·9-s + 4·11-s − 2·13-s − 2·17-s − 2·19-s − 8·23-s − 5·25-s − 8·31-s − 8·37-s + 6·41-s + 4·43-s + 8·47-s − 7·49-s − 12·53-s − 4·59-s − 2·61-s − 4·67-s − 4·71-s − 6·73-s − 8·79-s + 9·81-s + 12·83-s + 18·89-s − 6·97-s − 12·99-s + 101-s + 103-s + ⋯
L(s)  = 1  − 9-s + 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.458·19-s − 1.66·23-s − 25-s − 1.43·31-s − 1.31·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s − 49-s − 1.64·53-s − 0.520·59-s − 0.256·61-s − 0.488·67-s − 0.474·71-s − 0.702·73-s − 0.900·79-s + 81-s + 1.31·83-s + 1.90·89-s − 0.609·97-s − 1.20·99-s + 0.0995·101-s + 0.0985·103-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: \(0\)
Selberg data: \((2,\ 32192,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7575001951\)
\(L(\frac12)\) \(\approx\) \(0.7575001951\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−14.92568462801222, −14.41400390339989, −14.10194276228323, −13.68269647958567, −12.83077203664276, −12.33389734858885, −11.84804585988302, −11.44337329810221, −10.80374485411061, −10.31386505645730, −9.483521910025637, −9.159985381436000, −8.679920105711289, −7.876860471328038, −7.538344961731381, −6.692231328306062, −6.106397734653893, −5.786342032567607, −4.960035349050491, −4.190394982297831, −3.764926774274940, −2.995703155598720, −2.093947806378317, −1.671753504172076, −0.3112983675151166, 0.3112983675151166, 1.671753504172076, 2.093947806378317, 2.995703155598720, 3.764926774274940, 4.190394982297831, 4.960035349050491, 5.786342032567607, 6.106397734653893, 6.692231328306062, 7.538344961731381, 7.876860471328038, 8.679920105711289, 9.159985381436000, 9.483521910025637, 10.31386505645730, 10.80374485411061, 11.44337329810221, 11.84804585988302, 12.33389734858885, 12.83077203664276, 13.68269647958567, 14.10194276228323, 14.41400390339989, 14.92568462801222

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