Properties

Label 2-2e4-1.1-c13-0-5
Degree $2$
Conductor $16$
Sign $-1$
Analytic cond. $17.1569$
Root an. cond. $4.14209$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.35e3·3-s − 1.22e4·5-s − 4.52e5·7-s + 2.35e5·9-s − 7.28e6·11-s + 2.80e7·13-s − 1.65e7·15-s − 1.45e8·17-s − 1.08e8·19-s − 6.12e8·21-s − 1.59e8·23-s − 1.07e9·25-s − 1.83e9·27-s + 4.40e9·29-s − 5.53e9·31-s − 9.85e9·33-s + 5.53e9·35-s + 1.25e10·37-s + 3.79e10·39-s + 1.51e10·41-s + 3.15e10·43-s − 2.87e9·45-s − 3.50e10·47-s + 1.07e11·49-s − 1.96e11·51-s − 1.09e11·53-s + 8.90e10·55-s + ⋯
L(s)  = 1  + 1.07·3-s − 0.349·5-s − 1.45·7-s + 0.147·9-s − 1.24·11-s + 1.61·13-s − 0.374·15-s − 1.45·17-s − 0.528·19-s − 1.55·21-s − 0.225·23-s − 0.877·25-s − 0.913·27-s + 1.37·29-s − 1.12·31-s − 1.32·33-s + 0.508·35-s + 0.801·37-s + 1.72·39-s + 0.496·41-s + 0.759·43-s − 0.0515·45-s − 0.474·47-s + 1.11·49-s − 1.56·51-s − 0.681·53-s + 0.433·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-1$
Analytic conductor: \(17.1569\)
Root analytic conductor: \(4.14209\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 16,\ (\ :13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 - 1.35e3T + 1.59e6T^{2} \)
5 \( 1 + 1.22e4T + 1.22e9T^{2} \)
7 \( 1 + 4.52e5T + 9.68e10T^{2} \)
11 \( 1 + 7.28e6T + 3.45e13T^{2} \)
13 \( 1 - 2.80e7T + 3.02e14T^{2} \)
17 \( 1 + 1.45e8T + 9.90e15T^{2} \)
19 \( 1 + 1.08e8T + 4.20e16T^{2} \)
23 \( 1 + 1.59e8T + 5.04e17T^{2} \)
29 \( 1 - 4.40e9T + 1.02e19T^{2} \)
31 \( 1 + 5.53e9T + 2.44e19T^{2} \)
37 \( 1 - 1.25e10T + 2.43e20T^{2} \)
41 \( 1 - 1.51e10T + 9.25e20T^{2} \)
43 \( 1 - 3.15e10T + 1.71e21T^{2} \)
47 \( 1 + 3.50e10T + 5.46e21T^{2} \)
53 \( 1 + 1.09e11T + 2.60e22T^{2} \)
59 \( 1 - 1.18e11T + 1.04e23T^{2} \)
61 \( 1 + 2.98e11T + 1.61e23T^{2} \)
67 \( 1 + 7.44e11T + 5.48e23T^{2} \)
71 \( 1 - 1.15e12T + 1.16e24T^{2} \)
73 \( 1 - 1.84e12T + 1.67e24T^{2} \)
79 \( 1 - 2.20e12T + 4.66e24T^{2} \)
83 \( 1 - 1.92e12T + 8.87e24T^{2} \)
89 \( 1 + 5.50e12T + 2.19e25T^{2} \)
97 \( 1 - 1.35e13T + 6.73e25T^{2} \)
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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

−15.45320566557911739585848381208, −13.66595312993762257271194376020, −12.93589264550886955375061171692, −10.84838437184010403092606988067, −9.240502414505030205043998468535, −8.090087681447768409918649465184, −6.28027051356634941902753545627, −3.77846720030831692429756819543, −2.54482209674031040691983170860, 0, 2.54482209674031040691983170860, 3.77846720030831692429756819543, 6.28027051356634941902753545627, 8.090087681447768409918649465184, 9.240502414505030205043998468535, 10.84838437184010403092606988067, 12.93589264550886955375061171692, 13.66595312993762257271194376020, 15.45320566557911739585848381208

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