Properties

Label 2-2e4-1.1-c43-0-13
Degree $2$
Conductor $16$
Sign $-1$
Analytic cond. $187.376$
Root an. cond. $13.6885$
Motivic weight $43$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.22e10·3-s + 5.54e14·5-s + 5.57e17·7-s + 7.08e20·9-s + 1.13e22·11-s + 5.07e23·13-s − 1.78e25·15-s + 8.73e25·17-s − 5.55e27·19-s − 1.79e28·21-s − 1.85e29·23-s − 8.29e29·25-s − 1.22e31·27-s + 4.19e31·29-s + 9.30e31·31-s − 3.64e32·33-s + 3.09e32·35-s − 3.20e33·37-s − 1.63e34·39-s + 6.48e33·41-s − 3.51e33·43-s + 3.92e35·45-s − 6.21e35·47-s − 1.87e36·49-s − 2.81e36·51-s + 8.28e36·53-s + 6.26e36·55-s + ⋯
L(s)  = 1  − 1.77·3-s + 0.520·5-s + 0.377·7-s + 2.15·9-s + 0.460·11-s + 0.569·13-s − 0.924·15-s + 0.306·17-s − 1.78·19-s − 0.670·21-s − 0.978·23-s − 0.729·25-s − 2.06·27-s + 1.51·29-s + 0.802·31-s − 0.818·33-s + 0.196·35-s − 0.616·37-s − 1.01·39-s + 0.137·41-s − 0.0266·43-s + 1.12·45-s − 0.697·47-s − 0.857·49-s − 0.545·51-s + 0.702·53-s + 0.239·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(44-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+43/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-1$
Analytic conductor: \(187.376\)
Root analytic conductor: \(13.6885\)
Motivic weight: \(43\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 16,\ (\ :43/2),\ -1)\)

Particular Values

\(L(22)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{45}{2})\) 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 \( 1 + 3.22e10T + 3.28e20T^{2} \)
5 \( 1 - 5.54e14T + 1.13e30T^{2} \)
7 \( 1 - 5.57e17T + 2.18e36T^{2} \)
11 \( 1 - 1.13e22T + 6.02e44T^{2} \)
13 \( 1 - 5.07e23T + 7.93e47T^{2} \)
17 \( 1 - 8.73e25T + 8.11e52T^{2} \)
19 \( 1 + 5.55e27T + 9.69e54T^{2} \)
23 \( 1 + 1.85e29T + 3.58e58T^{2} \)
29 \( 1 - 4.19e31T + 7.64e62T^{2} \)
31 \( 1 - 9.30e31T + 1.34e64T^{2} \)
37 \( 1 + 3.20e33T + 2.70e67T^{2} \)
41 \( 1 - 6.48e33T + 2.23e69T^{2} \)
43 \( 1 + 3.51e33T + 1.73e70T^{2} \)
47 \( 1 + 6.21e35T + 7.94e71T^{2} \)
53 \( 1 - 8.28e36T + 1.39e74T^{2} \)
59 \( 1 - 7.48e36T + 1.40e76T^{2} \)
61 \( 1 + 2.24e38T + 5.87e76T^{2} \)
67 \( 1 - 1.89e39T + 3.32e78T^{2} \)
71 \( 1 - 9.53e39T + 4.01e79T^{2} \)
73 \( 1 - 2.48e39T + 1.32e80T^{2} \)
79 \( 1 - 3.50e39T + 3.96e81T^{2} \)
83 \( 1 - 2.23e41T + 3.31e82T^{2} \)
89 \( 1 + 4.38e41T + 6.66e83T^{2} \)
97 \( 1 + 6.11e42T + 2.69e85T^{2} \)
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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

−10.74488386325545104940898030439, −9.920022282412437513849268312572, −8.258145285186619947531062502293, −6.57186821723813724061768589912, −6.10614595687118437343557638714, −4.96175234453999793558738453284, −4.02079311618412851041077451945, −2.00147464839728397103899223670, −1.05165243470454079668918967295, 0, 1.05165243470454079668918967295, 2.00147464839728397103899223670, 4.02079311618412851041077451945, 4.96175234453999793558738453284, 6.10614595687118437343557638714, 6.57186821723813724061768589912, 8.258145285186619947531062502293, 9.920022282412437513849268312572, 10.74488386325545104940898030439

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