Properties

Label 2-2e4-1.1-c21-0-7
Degree $2$
Conductor $16$
Sign $-1$
Analytic cond. $44.7163$
Root an. cond. $6.68703$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.70e4·3-s − 3.39e7·5-s + 5.28e8·7-s − 1.03e9·9-s + 1.21e11·11-s + 4.33e11·13-s − 3.29e12·15-s − 1.31e13·17-s − 2.18e13·19-s + 5.13e13·21-s − 1.40e14·23-s + 6.74e14·25-s − 1.11e15·27-s + 1.17e15·29-s − 9.57e14·31-s + 1.18e16·33-s − 1.79e16·35-s − 3.53e16·37-s + 4.20e16·39-s − 1.71e17·41-s − 1.35e17·43-s + 3.51e16·45-s + 5.75e17·47-s − 2.78e17·49-s − 1.27e18·51-s − 9.62e17·53-s − 4.13e18·55-s + ⋯
L(s)  = 1  + 0.949·3-s − 1.55·5-s + 0.707·7-s − 0.0989·9-s + 1.41·11-s + 0.872·13-s − 1.47·15-s − 1.58·17-s − 0.818·19-s + 0.671·21-s − 0.708·23-s + 1.41·25-s − 1.04·27-s + 0.516·29-s − 0.209·31-s + 1.34·33-s − 1.09·35-s − 1.20·37-s + 0.828·39-s − 1.99·41-s − 0.957·43-s + 0.153·45-s + 1.59·47-s − 0.499·49-s − 1.50·51-s − 0.755·53-s − 2.20·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-1$
Analytic conductor: \(44.7163\)
Root analytic conductor: \(6.68703\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 16,\ (\ :21/2),\ -1)\)

Particular Values

\(L(11)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{23}{2})\) 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 \( 1 - 9.70e4T + 1.04e10T^{2} \)
5 \( 1 + 3.39e7T + 4.76e14T^{2} \)
7 \( 1 - 5.28e8T + 5.58e17T^{2} \)
11 \( 1 - 1.21e11T + 7.40e21T^{2} \)
13 \( 1 - 4.33e11T + 2.47e23T^{2} \)
17 \( 1 + 1.31e13T + 6.90e25T^{2} \)
19 \( 1 + 2.18e13T + 7.14e26T^{2} \)
23 \( 1 + 1.40e14T + 3.94e28T^{2} \)
29 \( 1 - 1.17e15T + 5.13e30T^{2} \)
31 \( 1 + 9.57e14T + 2.08e31T^{2} \)
37 \( 1 + 3.53e16T + 8.55e32T^{2} \)
41 \( 1 + 1.71e17T + 7.38e33T^{2} \)
43 \( 1 + 1.35e17T + 2.00e34T^{2} \)
47 \( 1 - 5.75e17T + 1.30e35T^{2} \)
53 \( 1 + 9.62e17T + 1.62e36T^{2} \)
59 \( 1 + 4.87e18T + 1.54e37T^{2} \)
61 \( 1 - 4.59e18T + 3.10e37T^{2} \)
67 \( 1 - 1.78e19T + 2.22e38T^{2} \)
71 \( 1 + 2.93e19T + 7.52e38T^{2} \)
73 \( 1 + 8.32e18T + 1.34e39T^{2} \)
79 \( 1 - 7.05e19T + 7.08e39T^{2} \)
83 \( 1 + 2.11e20T + 1.99e40T^{2} \)
89 \( 1 - 2.62e20T + 8.65e40T^{2} \)
97 \( 1 - 3.84e20T + 5.27e41T^{2} \)
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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

−13.75259437897908061459698863495, −11.95258652981533797251702989750, −11.07102716218618058281267418426, −8.780295142866150335947469084317, −8.278343143097965037197126763049, −6.73862480389650957085147133829, −4.33862925367507172476544966373, −3.51238623563548513343727677386, −1.75987227444394970102987374777, 0, 1.75987227444394970102987374777, 3.51238623563548513343727677386, 4.33862925367507172476544966373, 6.73862480389650957085147133829, 8.278343143097965037197126763049, 8.780295142866150335947469084317, 11.07102716218618058281267418426, 11.95258652981533797251702989750, 13.75259437897908061459698863495

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