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  − 1.96·3-s − 1.29·5-s − 5.21·7-s + 0.880·9-s − 5.81·11-s + 4.89·13-s + 2.55·15-s + 5.22·17-s − 19-s + 10.2·21-s + 4.72·23-s − 3.31·25-s + 4.17·27-s + 6.72·29-s + 3.44·31-s + 11.4·33-s + 6.76·35-s + 1.61·37-s − 9.63·39-s − 4.72·41-s − 1.03·43-s − 1.14·45-s + 1.95·47-s + 20.2·49-s − 10.2·51-s + 53-s + 7.54·55-s + ⋯
L(s)  = 1  − 1.13·3-s − 0.579·5-s − 1.97·7-s + 0.293·9-s − 1.75·11-s + 1.35·13-s + 0.659·15-s + 1.26·17-s − 0.229·19-s + 2.24·21-s + 0.985·23-s − 0.663·25-s + 0.803·27-s + 1.24·29-s + 0.618·31-s + 1.99·33-s + 1.14·35-s + 0.264·37-s − 1.54·39-s − 0.738·41-s − 0.157·43-s − 0.170·45-s + 0.285·47-s + 2.88·49-s − 1.44·51-s + 0.137·53-s + 1.01·55-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 \( 1 + 1.96T + 3T^{2} \)
5 \( 1 + 1.29T + 5T^{2} \)
7 \( 1 + 5.21T + 7T^{2} \)
11 \( 1 + 5.81T + 11T^{2} \)
13 \( 1 - 4.89T + 13T^{2} \)
17 \( 1 - 5.22T + 17T^{2} \)
23 \( 1 - 4.72T + 23T^{2} \)
29 \( 1 - 6.72T + 29T^{2} \)
31 \( 1 - 3.44T + 31T^{2} \)
37 \( 1 - 1.61T + 37T^{2} \)
41 \( 1 + 4.72T + 41T^{2} \)
43 \( 1 + 1.03T + 43T^{2} \)
47 \( 1 - 1.95T + 47T^{2} \)
59 \( 1 + 11.9T + 59T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 + 4.14T + 67T^{2} \)
71 \( 1 + 0.689T + 71T^{2} \)
73 \( 1 + 9.72T + 73T^{2} \)
79 \( 1 - 2.39T + 79T^{2} \)
83 \( 1 - 0.0104T + 83T^{2} \)
89 \( 1 + 8.42T + 89T^{2} \)
97 \( 1 + 12.3T + 97T^{2} \)
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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

−8.075621732708207718802355706830, −7.16107041574353148764352309720, −6.47426116967909488213872387844, −5.84622270951993273612501378210, −5.36297173782492128379624358637, −4.29801683017091447681615729615, −3.23177481895192636118933899549, −2.87898522712951977123692943532, −0.893562301369399154648904062134, 0, 0.893562301369399154648904062134, 2.87898522712951977123692943532, 3.23177481895192636118933899549, 4.29801683017091447681615729615, 5.36297173782492128379624358637, 5.84622270951993273612501378210, 6.47426116967909488213872387844, 7.16107041574353148764352309720, 8.075621732708207718802355706830

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