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  + 2.85·3-s − 0.0885·5-s − 1.03·7-s + 5.15·9-s − 3.21·11-s − 1.70·13-s − 0.253·15-s − 7.37·17-s − 19-s − 2.96·21-s − 8.03·23-s − 4.99·25-s + 6.16·27-s + 5.83·29-s + 4.90·31-s − 9.19·33-s + 0.0919·35-s − 4.56·37-s − 4.86·39-s + 4.12·41-s − 5.33·43-s − 0.457·45-s − 1.46·47-s − 5.92·49-s − 21.0·51-s + 53-s + 0.284·55-s + ⋯
L(s)  = 1  + 1.64·3-s − 0.0396·5-s − 0.392·7-s + 1.71·9-s − 0.970·11-s − 0.472·13-s − 0.0653·15-s − 1.78·17-s − 0.229·19-s − 0.646·21-s − 1.67·23-s − 0.998·25-s + 1.18·27-s + 1.08·29-s + 0.881·31-s − 1.59·33-s + 0.0155·35-s − 0.751·37-s − 0.779·39-s + 0.644·41-s − 0.813·43-s − 0.0681·45-s − 0.214·47-s − 0.846·49-s − 2.94·51-s + 0.137·53-s + 0.0384·55-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 \( 1 - 2.85T + 3T^{2} \)
5 \( 1 + 0.0885T + 5T^{2} \)
7 \( 1 + 1.03T + 7T^{2} \)
11 \( 1 + 3.21T + 11T^{2} \)
13 \( 1 + 1.70T + 13T^{2} \)
17 \( 1 + 7.37T + 17T^{2} \)
23 \( 1 + 8.03T + 23T^{2} \)
29 \( 1 - 5.83T + 29T^{2} \)
31 \( 1 - 4.90T + 31T^{2} \)
37 \( 1 + 4.56T + 37T^{2} \)
41 \( 1 - 4.12T + 41T^{2} \)
43 \( 1 + 5.33T + 43T^{2} \)
47 \( 1 + 1.46T + 47T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 + 9.61T + 61T^{2} \)
67 \( 1 - 4.16T + 67T^{2} \)
71 \( 1 + 16.1T + 71T^{2} \)
73 \( 1 - 7.73T + 73T^{2} \)
79 \( 1 - 1.03T + 79T^{2} \)
83 \( 1 - 2.95T + 83T^{2} \)
89 \( 1 + 3.28T + 89T^{2} \)
97 \( 1 + 4.36T + 97T^{2} \)
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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

−8.268884470098647340030955906268, −7.56183063314030250068199320544, −6.78288569372943993222025902448, −6.01169488502038124682037052158, −4.75361721381795659049793061406, −4.16215255579618689794848581870, −3.25380457914986638267596535947, −2.43998984957040531687053515648, −1.95567302675646714990980183304, 0, 1.95567302675646714990980183304, 2.43998984957040531687053515648, 3.25380457914986638267596535947, 4.16215255579618689794848581870, 4.75361721381795659049793061406, 6.01169488502038124682037052158, 6.78288569372943993222025902448, 7.56183063314030250068199320544, 8.268884470098647340030955906268

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