Properties

Degree 2
Conductor $ 2^{2} \cdot 19 \cdot 53 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.94·3-s + 1.42·5-s − 4.01·7-s + 0.801·9-s − 1.76·11-s − 0.162·13-s + 2.77·15-s + 3.69·17-s − 19-s − 7.83·21-s + 1.52·23-s − 2.97·25-s − 4.28·27-s + 2.66·29-s − 10.7·31-s − 3.44·33-s − 5.72·35-s + 1.29·37-s − 0.317·39-s − 7.80·41-s + 4.54·43-s + 1.14·45-s − 8.83·47-s + 9.14·49-s + 7.20·51-s + 53-s − 2.52·55-s + ⋯
L(s)  = 1  + 1.12·3-s + 0.636·5-s − 1.51·7-s + 0.267·9-s − 0.533·11-s − 0.0451·13-s + 0.717·15-s + 0.895·17-s − 0.229·19-s − 1.70·21-s + 0.318·23-s − 0.594·25-s − 0.824·27-s + 0.495·29-s − 1.93·31-s − 0.600·33-s − 0.967·35-s + 0.212·37-s − 0.0508·39-s − 1.21·41-s + 0.693·43-s + 0.170·45-s − 1.28·47-s + 1.30·49-s + 1.00·51-s + 0.137·53-s − 0.339·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4028\)    =    \(2^{2} \cdot 19 \cdot 53\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4028} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4028,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19,\;53\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;19,\;53\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
19 \( 1 + T \)
53 \( 1 - T \)
good3 \( 1 - 1.94T + 3T^{2} \)
5 \( 1 - 1.42T + 5T^{2} \)
7 \( 1 + 4.01T + 7T^{2} \)
11 \( 1 + 1.76T + 11T^{2} \)
13 \( 1 + 0.162T + 13T^{2} \)
17 \( 1 - 3.69T + 17T^{2} \)
23 \( 1 - 1.52T + 23T^{2} \)
29 \( 1 - 2.66T + 29T^{2} \)
31 \( 1 + 10.7T + 31T^{2} \)
37 \( 1 - 1.29T + 37T^{2} \)
41 \( 1 + 7.80T + 41T^{2} \)
43 \( 1 - 4.54T + 43T^{2} \)
47 \( 1 + 8.83T + 47T^{2} \)
59 \( 1 + 2.95T + 59T^{2} \)
61 \( 1 + 7.23T + 61T^{2} \)
67 \( 1 - 14.9T + 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 - 0.331T + 73T^{2} \)
79 \( 1 + 4.60T + 79T^{2} \)
83 \( 1 + 8.46T + 83T^{2} \)
89 \( 1 - 3.52T + 89T^{2} \)
97 \( 1 - 14.8T + 97T^{2} \)
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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

−8.105153188118747631528310111640, −7.45101180271261064857897597579, −6.64083876076409057092143302945, −5.87953943116034661101117879982, −5.23761055638978998618817796733, −3.89636083444496065615675756975, −3.25074123857291560636235362810, −2.66781640459346397822100615336, −1.71017924376825611071825026220, 0, 1.71017924376825611071825026220, 2.66781640459346397822100615336, 3.25074123857291560636235362810, 3.89636083444496065615675756975, 5.23761055638978998618817796733, 5.87953943116034661101117879982, 6.64083876076409057092143302945, 7.45101180271261064857897597579, 8.105153188118747631528310111640

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