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Properties

Label 2-32192-1.1-c1-0-18
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 + 2·5-s + 3·7-s + 6·9-s + 3·11-s − 3·13-s + 6·15-s + 2·17-s − 6·19-s + 9·21-s + 3·23-s − 25-s + 9·27-s − 2·29-s + 6·31-s + 9·33-s + 6·35-s + 2·37-s − 9·39-s + 10·41-s + 3·43-s + 12·45-s + 9·47-s + 2·49-s + 6·51-s − 2·53-s + 6·55-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.894·5-s + 1.13·7-s + 2·9-s + 0.904·11-s − 0.832·13-s + 1.54·15-s + 0.485·17-s − 1.37·19-s + 1.96·21-s + 0.625·23-s − 1/5·25-s + 1.73·27-s − 0.371·29-s + 1.07·31-s + 1.56·33-s + 1.01·35-s + 0.328·37-s − 1.44·39-s + 1.56·41-s + 0.457·43-s + 1.78·45-s + 1.31·47-s + 2/7·49-s + 0.840·51-s − 0.274·53-s + 0.809·55-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

\(L(1)\) \(\approx\) \(7.826723251\)
\(L(\frac12)\) \(\approx\) \(7.826723251\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
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

−14.86804121344734, −14.45536946425575, −14.11049922202370, −13.72431869614108, −13.09294450250483, −12.52490046120909, −12.07027848756501, −11.22181435358035, −10.66182069169113, −10.07480222458838, −9.407537318064856, −9.190151878954255, −8.634063991154309, −7.906276208814353, −7.707931000284983, −6.944934538609964, −6.270365473067815, −5.605565651499138, −4.632759967932890, −4.368537427051364, −3.622294160997204, −2.719534702020248, −2.280524575939834, −1.722629889842651, −1.016985951563992, 1.016985951563992, 1.722629889842651, 2.280524575939834, 2.719534702020248, 3.622294160997204, 4.368537427051364, 4.632759967932890, 5.605565651499138, 6.270365473067815, 6.944934538609964, 7.707931000284983, 7.906276208814353, 8.634063991154309, 9.190151878954255, 9.407537318064856, 10.07480222458838, 10.66182069169113, 11.22181435358035, 12.07027848756501, 12.52490046120909, 13.09294450250483, 13.72431869614108, 14.11049922202370, 14.45536946425575, 14.86804121344734

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