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  − 1.76·3-s − 1.36·5-s + 0.174·7-s + 0.119·9-s + 1.83·11-s + 3.53·13-s + 2.41·15-s − 6.42·17-s − 19-s − 0.307·21-s + 4.41·23-s − 3.12·25-s + 5.08·27-s + 0.386·29-s − 6.29·31-s − 3.24·33-s − 0.238·35-s + 7.89·37-s − 6.24·39-s − 1.09·41-s + 5.58·43-s − 0.163·45-s + 8.67·47-s − 6.96·49-s + 11.3·51-s + 53-s − 2.51·55-s + ⋯
L(s)  = 1  − 1.01·3-s − 0.612·5-s + 0.0658·7-s + 0.0397·9-s + 0.553·11-s + 0.981·13-s + 0.624·15-s − 1.55·17-s − 0.229·19-s − 0.0671·21-s + 0.920·23-s − 0.624·25-s + 0.979·27-s + 0.0717·29-s − 1.13·31-s − 0.564·33-s − 0.0403·35-s + 1.29·37-s − 1.00·39-s − 0.171·41-s + 0.851·43-s − 0.0243·45-s + 1.26·47-s − 0.995·49-s + 1.58·51-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 \( 1 + 1.76T + 3T^{2} \)
5 \( 1 + 1.36T + 5T^{2} \)
7 \( 1 - 0.174T + 7T^{2} \)
11 \( 1 - 1.83T + 11T^{2} \)
13 \( 1 - 3.53T + 13T^{2} \)
17 \( 1 + 6.42T + 17T^{2} \)
23 \( 1 - 4.41T + 23T^{2} \)
29 \( 1 - 0.386T + 29T^{2} \)
31 \( 1 + 6.29T + 31T^{2} \)
37 \( 1 - 7.89T + 37T^{2} \)
41 \( 1 + 1.09T + 41T^{2} \)
43 \( 1 - 5.58T + 43T^{2} \)
47 \( 1 - 8.67T + 47T^{2} \)
59 \( 1 + 10.0T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 + 12.8T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 - 2.20T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 - 14.5T + 97T^{2} \)
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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

−8.055163078380566889982145466165, −7.22966799732396108724236990115, −6.42867122526090328845104889272, −6.02203147196923295848586851866, −5.07746882310225651320134025419, −4.30371841325542575064410386184, −3.64488273149336947518520296212, −2.43438061178088378319407695413, −1.14235944175899039301871146016, 0, 1.14235944175899039301871146016, 2.43438061178088378319407695413, 3.64488273149336947518520296212, 4.30371841325542575064410386184, 5.07746882310225651320134025419, 6.02203147196923295848586851866, 6.42867122526090328845104889272, 7.22966799732396108724236990115, 8.055163078380566889982145466165

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