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Properties

Label 2-32192-1.1-c1-0-15
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-s + 4·5-s + 3·7-s − 2·9-s + 5·11-s − 13-s + 4·15-s + 8·19-s + 3·21-s − 9·23-s + 11·25-s − 5·27-s + 6·29-s + 2·31-s + 5·33-s + 12·35-s − 2·37-s − 39-s − 10·41-s + 5·43-s − 8·45-s + 47-s + 2·49-s + 6·53-s + 20·55-s + 8·57-s + 4·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.78·5-s + 1.13·7-s − 2/3·9-s + 1.50·11-s − 0.277·13-s + 1.03·15-s + 1.83·19-s + 0.654·21-s − 1.87·23-s + 11/5·25-s − 0.962·27-s + 1.11·29-s + 0.359·31-s + 0.870·33-s + 2.02·35-s − 0.328·37-s − 0.160·39-s − 1.56·41-s + 0.762·43-s − 1.19·45-s + 0.145·47-s + 2/7·49-s + 0.824·53-s + 2.69·55-s + 1.05·57-s + 0.520·59-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: \(0\)
Selberg data: \((2,\ 32192,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.050414631\)
\(L(\frac12)\) \(\approx\) \(6.050414631\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.64753877339685, −14.33940707330135, −14.03658271125386, −13.80503596385815, −13.25116930992537, −12.20764733837353, −11.87928746082983, −11.50293923188989, −10.67889348542944, −9.964248429434269, −9.741879267363860, −9.140832536122021, −8.597096434985770, −8.173147110109037, −7.418742773352598, −6.697190641722233, −6.151902854580463, −5.546583316229765, −5.152191140431104, −4.361306546879449, −3.563922047816846, −2.822138968941326, −2.110181848037567, −1.622989555962023, −0.9602182236189895, 0.9602182236189895, 1.622989555962023, 2.110181848037567, 2.822138968941326, 3.563922047816846, 4.361306546879449, 5.152191140431104, 5.546583316229765, 6.151902854580463, 6.697190641722233, 7.418742773352598, 8.173147110109037, 8.597096434985770, 9.140832536122021, 9.741879267363860, 9.964248429434269, 10.67889348542944, 11.50293923188989, 11.87928746082983, 12.20764733837353, 13.25116930992537, 13.80503596385815, 14.03658271125386, 14.33940707330135, 14.64753877339685

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