Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.39e7·3-s − 2.18e11·5-s − 4.92e13·7-s − 1.47e15·9-s − 2.23e17·11-s + 7.57e17·13-s + 1.39e19·15-s − 6.49e18·17-s − 7.92e20·19-s + 3.14e21·21-s + 2.99e22·23-s − 6.84e22·25-s + 4.49e23·27-s + 1.97e24·29-s + 4.54e24·31-s + 1.42e25·33-s + 1.07e25·35-s + 1.28e26·37-s − 4.84e25·39-s − 3.19e26·41-s − 1.35e27·43-s + 3.22e26·45-s − 3.49e27·47-s − 5.30e27·49-s + 4.15e26·51-s + 3.47e28·53-s + 4.88e28·55-s + ⋯
L(s)  = 1  − 0.857·3-s − 0.641·5-s − 0.560·7-s − 0.264·9-s − 1.46·11-s + 0.315·13-s + 0.550·15-s − 0.0323·17-s − 0.630·19-s + 0.480·21-s + 1.01·23-s − 0.588·25-s + 1.08·27-s + 1.46·29-s + 1.12·31-s + 1.25·33-s + 0.359·35-s + 1.70·37-s − 0.270·39-s − 0.782·41-s − 1.50·43-s + 0.169·45-s − 0.898·47-s − 0.686·49-s + 0.0277·51-s + 1.23·53-s + 0.939·55-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

\( d \)  =  \(2\)
\( N \)  =  \(4\)    =    \(2^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(33\)
character  :  $\chi_{4} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 4,\ (\ :33/2),\ 1)\)
\(L(17)\)  \(\approx\)  \(0.6332395951\)
\(L(\frac12)\)  \(\approx\)  \(0.6332395951\)
\(L(\frac{35}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 2$,\(F_p(T)\) is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 6.39e7T + 5.55e15T^{2} \)
5 \( 1 + 2.18e11T + 1.16e23T^{2} \)
7 \( 1 + 4.92e13T + 7.73e27T^{2} \)
11 \( 1 + 2.23e17T + 2.32e34T^{2} \)
13 \( 1 - 7.57e17T + 5.75e36T^{2} \)
17 \( 1 + 6.49e18T + 4.02e40T^{2} \)
19 \( 1 + 7.92e20T + 1.58e42T^{2} \)
23 \( 1 - 2.99e22T + 8.65e44T^{2} \)
29 \( 1 - 1.97e24T + 1.81e48T^{2} \)
31 \( 1 - 4.54e24T + 1.64e49T^{2} \)
37 \( 1 - 1.28e26T + 5.63e51T^{2} \)
41 \( 1 + 3.19e26T + 1.66e53T^{2} \)
43 \( 1 + 1.35e27T + 8.02e53T^{2} \)
47 \( 1 + 3.49e27T + 1.51e55T^{2} \)
53 \( 1 - 3.47e28T + 7.96e56T^{2} \)
59 \( 1 - 1.26e29T + 2.74e58T^{2} \)
61 \( 1 + 3.88e29T + 8.23e58T^{2} \)
67 \( 1 + 1.28e30T + 1.82e60T^{2} \)
71 \( 1 + 2.52e30T + 1.23e61T^{2} \)
73 \( 1 - 8.98e30T + 3.08e61T^{2} \)
79 \( 1 - 9.32e30T + 4.18e62T^{2} \)
83 \( 1 - 1.92e31T + 2.13e63T^{2} \)
89 \( 1 + 6.91e31T + 2.13e64T^{2} \)
97 \( 1 - 8.57e32T + 3.65e65T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.58536095215080620425668641927, −15.38972816538049612403309885469, −13.17541531217090719511557393482, −11.69519008438609794438957379778, −10.37787385145013433546168706059, −8.198090569384290375168761949202, −6.38607836330697214818893296773, −4.86824427702834250115861835213, −2.92070769047758086335416690277, −0.50970554940105003785325212622, 0.50970554940105003785325212622, 2.92070769047758086335416690277, 4.86824427702834250115861835213, 6.38607836330697214818893296773, 8.198090569384290375168761949202, 10.37787385145013433546168706059, 11.69519008438609794438957379778, 13.17541531217090719511557393482, 15.38972816538049612403309885469, 16.58536095215080620425668641927

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