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  − 2.05·3-s + 2.67·5-s + 0.688·7-s + 1.20·9-s − 5.36·11-s + 1.75·13-s − 5.48·15-s + 3.58·17-s − 19-s − 1.41·21-s + 1.71·23-s + 2.14·25-s + 3.67·27-s − 10.2·29-s − 1.27·31-s + 11.0·33-s + 1.84·35-s + 1.26·37-s − 3.59·39-s + 9.70·41-s + 2.83·43-s + 3.23·45-s − 7.30·47-s − 6.52·49-s − 7.35·51-s + 53-s − 14.3·55-s + ⋯
L(s)  = 1  − 1.18·3-s + 1.19·5-s + 0.260·7-s + 0.403·9-s − 1.61·11-s + 0.485·13-s − 1.41·15-s + 0.868·17-s − 0.229·19-s − 0.308·21-s + 0.356·23-s + 0.429·25-s + 0.706·27-s − 1.90·29-s − 0.228·31-s + 1.91·33-s + 0.311·35-s + 0.207·37-s − 0.575·39-s + 1.51·41-s + 0.431·43-s + 0.481·45-s − 1.06·47-s − 0.932·49-s − 1.02·51-s + 0.137·53-s − 1.93·55-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 \( 1 + 2.05T + 3T^{2} \)
5 \( 1 - 2.67T + 5T^{2} \)
7 \( 1 - 0.688T + 7T^{2} \)
11 \( 1 + 5.36T + 11T^{2} \)
13 \( 1 - 1.75T + 13T^{2} \)
17 \( 1 - 3.58T + 17T^{2} \)
23 \( 1 - 1.71T + 23T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
31 \( 1 + 1.27T + 31T^{2} \)
37 \( 1 - 1.26T + 37T^{2} \)
41 \( 1 - 9.70T + 41T^{2} \)
43 \( 1 - 2.83T + 43T^{2} \)
47 \( 1 + 7.30T + 47T^{2} \)
59 \( 1 + 7.26T + 59T^{2} \)
61 \( 1 + 14.4T + 61T^{2} \)
67 \( 1 + 4.38T + 67T^{2} \)
71 \( 1 + 0.131T + 71T^{2} \)
73 \( 1 - 9.46T + 73T^{2} \)
79 \( 1 + 7.19T + 79T^{2} \)
83 \( 1 + 6.22T + 83T^{2} \)
89 \( 1 - 6.04T + 89T^{2} \)
97 \( 1 - 10.7T + 97T^{2} \)
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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

−7.88818729947988447463111148906, −7.38600227034817870540338162296, −6.15931205228343363418281581245, −5.89585541075429373029162548599, −5.27348631131377505741361327037, −4.67247704099795783618592633817, −3.31552242451528130257445798067, −2.33477307279299007602425741658, −1.35011421379858675837205741532, 0, 1.35011421379858675837205741532, 2.33477307279299007602425741658, 3.31552242451528130257445798067, 4.67247704099795783618592633817, 5.27348631131377505741361327037, 5.89585541075429373029162548599, 6.15931205228343363418281581245, 7.38600227034817870540338162296, 7.88818729947988447463111148906

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