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  + 0.755·3-s + 4.26·5-s − 2.09·7-s − 2.42·9-s − 4.58·11-s + 1.90·13-s + 3.22·15-s − 5.40·17-s − 19-s − 1.58·21-s − 4.46·23-s + 13.2·25-s − 4.10·27-s + 1.04·29-s − 6.77·31-s − 3.46·33-s − 8.92·35-s − 1.41·37-s + 1.44·39-s + 4.93·41-s + 1.29·43-s − 10.3·45-s + 8.54·47-s − 2.62·49-s − 4.08·51-s + 53-s − 19.5·55-s + ⋯
L(s)  = 1  + 0.436·3-s + 1.90·5-s − 0.790·7-s − 0.809·9-s − 1.38·11-s + 0.528·13-s + 0.833·15-s − 1.31·17-s − 0.229·19-s − 0.344·21-s − 0.931·23-s + 2.64·25-s − 0.789·27-s + 0.193·29-s − 1.21·31-s − 0.602·33-s − 1.50·35-s − 0.232·37-s + 0.230·39-s + 0.770·41-s + 0.197·43-s − 1.54·45-s + 1.24·47-s − 0.375·49-s − 0.572·51-s + 0.137·53-s − 2.63·55-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 \( 1 - 0.755T + 3T^{2} \)
5 \( 1 - 4.26T + 5T^{2} \)
7 \( 1 + 2.09T + 7T^{2} \)
11 \( 1 + 4.58T + 11T^{2} \)
13 \( 1 - 1.90T + 13T^{2} \)
17 \( 1 + 5.40T + 17T^{2} \)
23 \( 1 + 4.46T + 23T^{2} \)
29 \( 1 - 1.04T + 29T^{2} \)
31 \( 1 + 6.77T + 31T^{2} \)
37 \( 1 + 1.41T + 37T^{2} \)
41 \( 1 - 4.93T + 41T^{2} \)
43 \( 1 - 1.29T + 43T^{2} \)
47 \( 1 - 8.54T + 47T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 1.99T + 61T^{2} \)
67 \( 1 + 9.79T + 67T^{2} \)
71 \( 1 - 4.71T + 71T^{2} \)
73 \( 1 + 6.64T + 73T^{2} \)
79 \( 1 + 5.80T + 79T^{2} \)
83 \( 1 + 14.1T + 83T^{2} \)
89 \( 1 + 12.5T + 89T^{2} \)
97 \( 1 + 9.06T + 97T^{2} \)
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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

−8.290744080979944379325901619349, −7.27778846444547897677390672125, −6.39653250608277548350511350718, −5.83130974702548721042896159616, −5.41192018481290903983214050277, −4.27983585158325744533311283260, −3.00830826213383015103800052942, −2.51842149490043962200220957807, −1.77232324285410911951801193045, 0, 1.77232324285410911951801193045, 2.51842149490043962200220957807, 3.00830826213383015103800052942, 4.27983585158325744533311283260, 5.41192018481290903983214050277, 5.83130974702548721042896159616, 6.39653250608277548350511350718, 7.27778846444547897677390672125, 8.290744080979944379325901619349

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