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Properties

Label 2-32192-1.1-c1-0-5
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  + 3·3-s − 3·7-s + 6·9-s − 11-s − 5·13-s + 4·17-s + 8·19-s − 9·21-s − 7·23-s − 5·25-s + 9·27-s + 6·29-s + 2·31-s − 3·33-s + 10·37-s − 15·39-s − 6·41-s − 43-s + 7·47-s + 2·49-s + 12·51-s − 6·53-s + 24·57-s + 4·59-s + 7·61-s − 18·63-s + 7·67-s + ⋯
L(s)  = 1  + 1.73·3-s − 1.13·7-s + 2·9-s − 0.301·11-s − 1.38·13-s + 0.970·17-s + 1.83·19-s − 1.96·21-s − 1.45·23-s − 25-s + 1.73·27-s + 1.11·29-s + 0.359·31-s − 0.522·33-s + 1.64·37-s − 2.40·39-s − 0.937·41-s − 0.152·43-s + 1.02·47-s + 2/7·49-s + 1.68·51-s − 0.824·53-s + 3.17·57-s + 0.520·59-s + 0.896·61-s − 2.26·63-s + 0.855·67-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

\(L(1)\) \(\approx\) \(3.717151330\)
\(L(\frac12)\) \(\approx\) \(3.717151330\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−14.92967737928214, −14.43676837644126, −13.98439750005585, −13.64158863659318, −13.10975524941308, −12.44508480629507, −12.06920166371599, −11.56013680335483, −10.30405883109588, −9.944087755587962, −9.664718645611243, −9.375309723617791, −8.444633333584086, −7.887609359613539, −7.629018746386264, −7.047529486800738, −6.309683095867102, −5.573943155832338, −4.863204647380540, −4.018906859539149, −3.535112765352664, −2.812376473694685, −2.605004048822736, −1.686196828224011, −0.6426129362884368, 0.6426129362884368, 1.686196828224011, 2.605004048822736, 2.812376473694685, 3.535112765352664, 4.018906859539149, 4.863204647380540, 5.573943155832338, 6.309683095867102, 7.047529486800738, 7.629018746386264, 7.887609359613539, 8.444633333584086, 9.375309723617791, 9.664718645611243, 9.944087755587962, 10.30405883109588, 11.56013680335483, 12.06920166371599, 12.44508480629507, 13.10975524941308, 13.64158863659318, 13.98439750005585, 14.43676837644126, 14.92967737928214

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