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  − 0.383·3-s − 4.10·5-s + 0.287·7-s − 2.85·9-s + 0.513·11-s + 2.10·13-s + 1.57·15-s − 1.15·17-s − 19-s − 0.110·21-s + 8.21·23-s + 11.8·25-s + 2.24·27-s − 0.842·29-s + 5.84·31-s − 0.197·33-s − 1.17·35-s − 4.24·37-s − 0.809·39-s + 3.80·41-s − 1.27·43-s + 11.7·45-s + 5.24·47-s − 6.91·49-s + 0.443·51-s + 53-s − 2.10·55-s + ⋯
L(s)  = 1  − 0.221·3-s − 1.83·5-s + 0.108·7-s − 0.950·9-s + 0.154·11-s + 0.584·13-s + 0.406·15-s − 0.279·17-s − 0.229·19-s − 0.0240·21-s + 1.71·23-s + 2.36·25-s + 0.432·27-s − 0.156·29-s + 1.04·31-s − 0.0343·33-s − 0.199·35-s − 0.698·37-s − 0.129·39-s + 0.594·41-s − 0.194·43-s + 1.74·45-s + 0.765·47-s − 0.988·49-s + 0.0620·51-s + 0.137·53-s − 0.284·55-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 \( 1 + 0.383T + 3T^{2} \)
5 \( 1 + 4.10T + 5T^{2} \)
7 \( 1 - 0.287T + 7T^{2} \)
11 \( 1 - 0.513T + 11T^{2} \)
13 \( 1 - 2.10T + 13T^{2} \)
17 \( 1 + 1.15T + 17T^{2} \)
23 \( 1 - 8.21T + 23T^{2} \)
29 \( 1 + 0.842T + 29T^{2} \)
31 \( 1 - 5.84T + 31T^{2} \)
37 \( 1 + 4.24T + 37T^{2} \)
41 \( 1 - 3.80T + 41T^{2} \)
43 \( 1 + 1.27T + 43T^{2} \)
47 \( 1 - 5.24T + 47T^{2} \)
59 \( 1 - 3.73T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 + 0.286T + 67T^{2} \)
71 \( 1 - 3.50T + 71T^{2} \)
73 \( 1 + 7.34T + 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 + 9.32T + 83T^{2} \)
89 \( 1 - 2.32T + 89T^{2} \)
97 \( 1 + 17.8T + 97T^{2} \)
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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

−8.207042397931381800037566640185, −7.34502426704659470720896681100, −6.76998659543215347754493846332, −5.87169473017293860150098771160, −4.89175091894905509832236871469, −4.30013340416860122225690225917, −3.38720698447819154882616364444, −2.80280424569598013711284870959, −1.09639042566798634078494325829, 0, 1.09639042566798634078494325829, 2.80280424569598013711284870959, 3.38720698447819154882616364444, 4.30013340416860122225690225917, 4.89175091894905509832236871469, 5.87169473017293860150098771160, 6.76998659543215347754493846332, 7.34502426704659470720896681100, 8.207042397931381800037566640185

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