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  + 1.67·3-s − 0.778·5-s + 3.24·7-s − 0.201·9-s − 5.19·11-s − 4.82·13-s − 1.30·15-s + 1.25·17-s − 19-s + 5.42·21-s + 8.02·23-s − 4.39·25-s − 5.35·27-s − 3.73·29-s + 8.22·31-s − 8.68·33-s − 2.52·35-s − 4.60·37-s − 8.07·39-s − 7.33·41-s − 7.17·43-s + 0.156·45-s + 2.69·47-s + 3.53·49-s + 2.09·51-s + 53-s + 4.04·55-s + ⋯
L(s)  = 1  + 0.965·3-s − 0.348·5-s + 1.22·7-s − 0.0671·9-s − 1.56·11-s − 1.33·13-s − 0.336·15-s + 0.303·17-s − 0.229·19-s + 1.18·21-s + 1.67·23-s − 0.878·25-s − 1.03·27-s − 0.693·29-s + 1.47·31-s − 1.51·33-s − 0.426·35-s − 0.757·37-s − 1.29·39-s − 1.14·41-s − 1.09·43-s + 0.0233·45-s + 0.392·47-s + 0.504·49-s + 0.293·51-s + 0.137·53-s + 0.545·55-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 \( 1 - 1.67T + 3T^{2} \)
5 \( 1 + 0.778T + 5T^{2} \)
7 \( 1 - 3.24T + 7T^{2} \)
11 \( 1 + 5.19T + 11T^{2} \)
13 \( 1 + 4.82T + 13T^{2} \)
17 \( 1 - 1.25T + 17T^{2} \)
23 \( 1 - 8.02T + 23T^{2} \)
29 \( 1 + 3.73T + 29T^{2} \)
31 \( 1 - 8.22T + 31T^{2} \)
37 \( 1 + 4.60T + 37T^{2} \)
41 \( 1 + 7.33T + 41T^{2} \)
43 \( 1 + 7.17T + 43T^{2} \)
47 \( 1 - 2.69T + 47T^{2} \)
59 \( 1 - 3.81T + 59T^{2} \)
61 \( 1 + 2.56T + 61T^{2} \)
67 \( 1 + 6.53T + 67T^{2} \)
71 \( 1 - 5.85T + 71T^{2} \)
73 \( 1 + 11.1T + 73T^{2} \)
79 \( 1 + 2.01T + 79T^{2} \)
83 \( 1 + 8.37T + 83T^{2} \)
89 \( 1 + 14.3T + 89T^{2} \)
97 \( 1 - 8.42T + 97T^{2} \)
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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

−8.214338506071089551787408884016, −7.53737618245583612131567549625, −6.99692198084153288145720082127, −5.51385229987746975676808641586, −5.08450406471931880014120882152, −4.33322868140592553051449552393, −3.13140886953547584305951257600, −2.59890961805298718911393814692, −1.69689576641308101280967977533, 0, 1.69689576641308101280967977533, 2.59890961805298718911393814692, 3.13140886953547584305951257600, 4.33322868140592553051449552393, 5.08450406471931880014120882152, 5.51385229987746975676808641586, 6.99692198084153288145720082127, 7.53737618245583612131567549625, 8.214338506071089551787408884016

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