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  + 0.346·3-s + 0.720·5-s − 2.22·7-s − 2.88·9-s + 2.78·11-s + 3.24·13-s + 0.249·15-s − 0.710·17-s − 19-s − 0.769·21-s + 1.68·23-s − 4.48·25-s − 2.03·27-s − 7.75·29-s + 1.25·31-s + 0.964·33-s − 1.60·35-s + 2.09·37-s + 1.12·39-s − 2.36·41-s − 10.0·43-s − 2.07·45-s + 0.417·47-s − 2.06·49-s − 0.246·51-s + 53-s + 2.00·55-s + ⋯
L(s)  = 1  + 0.199·3-s + 0.322·5-s − 0.839·7-s − 0.960·9-s + 0.839·11-s + 0.899·13-s + 0.0644·15-s − 0.172·17-s − 0.229·19-s − 0.167·21-s + 0.351·23-s − 0.896·25-s − 0.391·27-s − 1.43·29-s + 0.226·31-s + 0.167·33-s − 0.270·35-s + 0.345·37-s + 0.179·39-s − 0.369·41-s − 1.53·43-s − 0.309·45-s + 0.0608·47-s − 0.294·49-s − 0.0344·51-s + 0.137·53-s + 0.270·55-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 \( 1 - 0.346T + 3T^{2} \)
5 \( 1 - 0.720T + 5T^{2} \)
7 \( 1 + 2.22T + 7T^{2} \)
11 \( 1 - 2.78T + 11T^{2} \)
13 \( 1 - 3.24T + 13T^{2} \)
17 \( 1 + 0.710T + 17T^{2} \)
23 \( 1 - 1.68T + 23T^{2} \)
29 \( 1 + 7.75T + 29T^{2} \)
31 \( 1 - 1.25T + 31T^{2} \)
37 \( 1 - 2.09T + 37T^{2} \)
41 \( 1 + 2.36T + 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 - 0.417T + 47T^{2} \)
59 \( 1 - 3.65T + 59T^{2} \)
61 \( 1 - 13.5T + 61T^{2} \)
67 \( 1 - 8.76T + 67T^{2} \)
71 \( 1 + 2.98T + 71T^{2} \)
73 \( 1 + 6.93T + 73T^{2} \)
79 \( 1 + 8.08T + 79T^{2} \)
83 \( 1 + 5.16T + 83T^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 + 7.60T + 97T^{2} \)
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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

−8.376193646991748001993390884835, −7.23398843049194567721801753422, −6.50706117688526198260718833013, −5.94957797183640207577413474964, −5.27223250304531851797569848039, −3.97688007013447433110107621217, −3.48932413834635978153402078454, −2.53360575068645668735105183990, −1.48011405788428173584104020483, 0, 1.48011405788428173584104020483, 2.53360575068645668735105183990, 3.48932413834635978153402078454, 3.97688007013447433110107621217, 5.27223250304531851797569848039, 5.94957797183640207577413474964, 6.50706117688526198260718833013, 7.23398843049194567721801753422, 8.376193646991748001993390884835

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