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  + 2.16e7·3-s + 6.04e11·5-s + 1.12e14·7-s − 5.08e15·9-s + 2.12e17·11-s − 2.20e18·13-s + 1.31e19·15-s − 9.09e19·17-s + 5.28e20·19-s + 2.43e21·21-s − 2.52e22·23-s + 2.49e23·25-s − 2.31e23·27-s + 2.22e24·29-s + 5.96e24·31-s + 4.59e24·33-s + 6.78e25·35-s − 1.07e25·37-s − 4.77e25·39-s + 5.45e25·41-s − 7.34e26·43-s − 3.07e27·45-s − 2.88e27·47-s + 4.86e27·49-s − 1.97e27·51-s − 6.58e27·53-s + 1.28e29·55-s + ⋯
L(s)  = 1  + 0.290·3-s + 1.77·5-s + 1.27·7-s − 0.915·9-s + 1.39·11-s − 0.917·13-s + 0.515·15-s − 0.453·17-s + 0.420·19-s + 0.371·21-s − 0.858·23-s + 2.14·25-s − 0.557·27-s + 1.65·29-s + 1.47·31-s + 0.404·33-s + 2.26·35-s − 0.143·37-s − 0.266·39-s + 0.133·41-s − 0.820·43-s − 1.62·45-s − 0.741·47-s + 0.629·49-s − 0.131·51-s − 0.233·53-s + 2.46·55-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

\(L(17)\) \(\approx\) \(3.382012769\)
\(L(\frac12)\) \(\approx\) \(3.382012769\)
\(L(\frac{35}{2})\) 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 \( 1 - 2.16e7T + 5.55e15T^{2} \)
5 \( 1 - 6.04e11T + 1.16e23T^{2} \)
7 \( 1 - 1.12e14T + 7.73e27T^{2} \)
11 \( 1 - 2.12e17T + 2.32e34T^{2} \)
13 \( 1 + 2.20e18T + 5.75e36T^{2} \)
17 \( 1 + 9.09e19T + 4.02e40T^{2} \)
19 \( 1 - 5.28e20T + 1.58e42T^{2} \)
23 \( 1 + 2.52e22T + 8.65e44T^{2} \)
29 \( 1 - 2.22e24T + 1.81e48T^{2} \)
31 \( 1 - 5.96e24T + 1.64e49T^{2} \)
37 \( 1 + 1.07e25T + 5.63e51T^{2} \)
41 \( 1 - 5.45e25T + 1.66e53T^{2} \)
43 \( 1 + 7.34e26T + 8.02e53T^{2} \)
47 \( 1 + 2.88e27T + 1.51e55T^{2} \)
53 \( 1 + 6.58e27T + 7.96e56T^{2} \)
59 \( 1 + 1.55e29T + 2.74e58T^{2} \)
61 \( 1 - 5.35e29T + 8.23e58T^{2} \)
67 \( 1 + 5.20e29T + 1.82e60T^{2} \)
71 \( 1 - 2.97e30T + 1.23e61T^{2} \)
73 \( 1 + 5.09e30T + 3.08e61T^{2} \)
79 \( 1 + 1.81e30T + 4.18e62T^{2} \)
83 \( 1 + 3.91e31T + 2.13e63T^{2} \)
89 \( 1 + 1.76e32T + 2.13e64T^{2} \)
97 \( 1 - 8.13e32T + 3.65e65T^{2} \)
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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

−17.19705859177165266343889893123, −14.46332418152513106338421879764, −13.93535191001986867513127557801, −11.73590723627168521649275674842, −9.890616235970197654756947968842, −8.549824988887918785174758742834, −6.34431902832542930260175433855, −4.90104549828883922952571876861, −2.48515890102930602818381678793, −1.36732755296961624989494891612, 1.36732755296961624989494891612, 2.48515890102930602818381678793, 4.90104549828883922952571876861, 6.34431902832542930260175433855, 8.549824988887918785174758742834, 9.890616235970197654756947968842, 11.73590723627168521649275674842, 13.93535191001986867513127557801, 14.46332418152513106338421879764, 17.19705859177165266343889893123

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