Properties

Degree $2$
Conductor $4$
Sign $1$
Motivic weight $33$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.34e8·3-s − 4.39e11·5-s − 5.84e13·7-s + 1.25e16·9-s + 2.38e17·11-s + 1.71e18·13-s − 5.92e19·15-s + 1.90e20·17-s + 1.58e21·19-s − 7.87e21·21-s + 6.96e21·23-s + 7.70e22·25-s + 9.47e23·27-s − 1.30e24·29-s + 5.31e24·31-s + 3.21e25·33-s + 2.57e25·35-s − 8.25e25·37-s + 2.31e26·39-s − 4.09e26·41-s + 5.28e26·43-s − 5.53e27·45-s − 8.51e26·47-s − 4.31e27·49-s + 2.56e28·51-s + 2.58e28·53-s − 1.04e29·55-s + ⋯
L(s)  = 1  + 1.80·3-s − 1.28·5-s − 0.664·7-s + 2.26·9-s + 1.56·11-s + 0.715·13-s − 2.32·15-s + 0.949·17-s + 1.26·19-s − 1.20·21-s + 0.236·23-s + 0.662·25-s + 2.28·27-s − 0.969·29-s + 1.31·31-s + 2.83·33-s + 0.856·35-s − 1.09·37-s + 1.29·39-s − 1.00·41-s + 0.590·43-s − 2.92·45-s − 0.218·47-s − 0.558·49-s + 1.71·51-s + 0.917·53-s − 2.01·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $1$
Motivic weight: \(33\)
Character: $\chi_{4} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :33/2),\ 1)\)

Particular Values

\(L(17)\) \(\approx\) \(3.320755470\)
\(L(\frac12)\) \(\approx\) \(3.320755470\)
\(L(\frac{35}{2})\) 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 \( 1 - 1.34e8T + 5.55e15T^{2} \)
5 \( 1 + 4.39e11T + 1.16e23T^{2} \)
7 \( 1 + 5.84e13T + 7.73e27T^{2} \)
11 \( 1 - 2.38e17T + 2.32e34T^{2} \)
13 \( 1 - 1.71e18T + 5.75e36T^{2} \)
17 \( 1 - 1.90e20T + 4.02e40T^{2} \)
19 \( 1 - 1.58e21T + 1.58e42T^{2} \)
23 \( 1 - 6.96e21T + 8.65e44T^{2} \)
29 \( 1 + 1.30e24T + 1.81e48T^{2} \)
31 \( 1 - 5.31e24T + 1.64e49T^{2} \)
37 \( 1 + 8.25e25T + 5.63e51T^{2} \)
41 \( 1 + 4.09e26T + 1.66e53T^{2} \)
43 \( 1 - 5.28e26T + 8.02e53T^{2} \)
47 \( 1 + 8.51e26T + 1.51e55T^{2} \)
53 \( 1 - 2.58e28T + 7.96e56T^{2} \)
59 \( 1 - 2.29e29T + 2.74e58T^{2} \)
61 \( 1 + 8.85e28T + 8.23e58T^{2} \)
67 \( 1 + 1.09e30T + 1.82e60T^{2} \)
71 \( 1 - 1.24e30T + 1.23e61T^{2} \)
73 \( 1 + 2.58e30T + 3.08e61T^{2} \)
79 \( 1 - 3.55e31T + 4.18e62T^{2} \)
83 \( 1 + 5.14e31T + 2.13e63T^{2} \)
89 \( 1 - 5.16e31T + 2.13e64T^{2} \)
97 \( 1 + 1.10e33T + 3.65e65T^{2} \)
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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

−16.12647252471748147710283249831, −14.96879042088726446701295147087, −13.72704328342322776545814465675, −11.99025560246433380318376728905, −9.567785108082663772776959964757, −8.362685900156924478337277592799, −7.09976330916847255445578417588, −3.86237457259214702047208929759, −3.26949025307006799163845041189, −1.21308022921594800971515460596, 1.21308022921594800971515460596, 3.26949025307006799163845041189, 3.86237457259214702047208929759, 7.09976330916847255445578417588, 8.362685900156924478337277592799, 9.567785108082663772776959964757, 11.99025560246433380318376728905, 13.72704328342322776545814465675, 14.96879042088726446701295147087, 16.12647252471748147710283249831

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