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  − 2.40·3-s − 0.489·5-s + 4.26·7-s + 2.78·9-s − 0.593·11-s − 1.15·13-s + 1.17·15-s − 0.567·17-s − 19-s − 10.2·21-s − 0.283·23-s − 4.76·25-s + 0.524·27-s + 7.84·29-s − 1.45·31-s + 1.42·33-s − 2.08·35-s − 10.2·37-s + 2.78·39-s − 1.80·41-s − 10.9·43-s − 1.36·45-s − 0.268·47-s + 11.2·49-s + 1.36·51-s + 53-s + 0.290·55-s + ⋯
L(s)  = 1  − 1.38·3-s − 0.218·5-s + 1.61·7-s + 0.927·9-s − 0.178·11-s − 0.320·13-s + 0.303·15-s − 0.137·17-s − 0.229·19-s − 2.23·21-s − 0.0591·23-s − 0.952·25-s + 0.100·27-s + 1.45·29-s − 0.261·31-s + 0.248·33-s − 0.353·35-s − 1.67·37-s + 0.445·39-s − 0.281·41-s − 1.67·43-s − 0.203·45-s − 0.0391·47-s + 1.60·49-s + 0.190·51-s + 0.137·53-s + 0.0391·55-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 \( 1 + 2.40T + 3T^{2} \)
5 \( 1 + 0.489T + 5T^{2} \)
7 \( 1 - 4.26T + 7T^{2} \)
11 \( 1 + 0.593T + 11T^{2} \)
13 \( 1 + 1.15T + 13T^{2} \)
17 \( 1 + 0.567T + 17T^{2} \)
23 \( 1 + 0.283T + 23T^{2} \)
29 \( 1 - 7.84T + 29T^{2} \)
31 \( 1 + 1.45T + 31T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 + 1.80T + 41T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 + 0.268T + 47T^{2} \)
59 \( 1 + 7.12T + 59T^{2} \)
61 \( 1 + 3.60T + 61T^{2} \)
67 \( 1 - 13.0T + 67T^{2} \)
71 \( 1 - 5.30T + 71T^{2} \)
73 \( 1 - 6.82T + 73T^{2} \)
79 \( 1 - 6.03T + 79T^{2} \)
83 \( 1 + 18.1T + 83T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 - 16.7T + 97T^{2} \)
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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

−8.163421863775715822428691334193, −7.23331750615098349422188568774, −6.57560561330334415451609152596, −5.72080239111543101292733169210, −4.92877650504947778623159743379, −4.75531606576294591630264824393, −3.62010110584056019368499300450, −2.19531279571354279761448785681, −1.27545318359038631543091908218, 0, 1.27545318359038631543091908218, 2.19531279571354279761448785681, 3.62010110584056019368499300450, 4.75531606576294591630264824393, 4.92877650504947778623159743379, 5.72080239111543101292733169210, 6.57560561330334415451609152596, 7.23331750615098349422188568774, 8.163421863775715822428691334193

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