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  + 0.537·3-s − 2.35·5-s − 1.00·7-s − 2.71·9-s + 3.55·11-s + 0.172·13-s − 1.26·15-s + 7.68·17-s − 19-s − 0.542·21-s − 8.17·23-s + 0.531·25-s − 3.06·27-s + 3.95·29-s + 8.96·31-s + 1.90·33-s + 2.37·35-s − 9.28·37-s + 0.0924·39-s − 0.449·41-s + 9.15·43-s + 6.37·45-s − 9.50·47-s − 5.98·49-s + 4.13·51-s + 53-s − 8.35·55-s + ⋯
L(s)  = 1  + 0.310·3-s − 1.05·5-s − 0.381·7-s − 0.903·9-s + 1.07·11-s + 0.0477·13-s − 0.326·15-s + 1.86·17-s − 0.229·19-s − 0.118·21-s − 1.70·23-s + 0.106·25-s − 0.590·27-s + 0.735·29-s + 1.61·31-s + 0.332·33-s + 0.401·35-s − 1.52·37-s + 0.0148·39-s − 0.0702·41-s + 1.39·43-s + 0.950·45-s − 1.38·47-s − 0.854·49-s + 0.578·51-s + 0.137·53-s − 1.12·55-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 \( 1 - 0.537T + 3T^{2} \)
5 \( 1 + 2.35T + 5T^{2} \)
7 \( 1 + 1.00T + 7T^{2} \)
11 \( 1 - 3.55T + 11T^{2} \)
13 \( 1 - 0.172T + 13T^{2} \)
17 \( 1 - 7.68T + 17T^{2} \)
23 \( 1 + 8.17T + 23T^{2} \)
29 \( 1 - 3.95T + 29T^{2} \)
31 \( 1 - 8.96T + 31T^{2} \)
37 \( 1 + 9.28T + 37T^{2} \)
41 \( 1 + 0.449T + 41T^{2} \)
43 \( 1 - 9.15T + 43T^{2} \)
47 \( 1 + 9.50T + 47T^{2} \)
59 \( 1 + 7.97T + 59T^{2} \)
61 \( 1 - 3.34T + 61T^{2} \)
67 \( 1 + 4.16T + 67T^{2} \)
71 \( 1 - 4.34T + 71T^{2} \)
73 \( 1 - 3.25T + 73T^{2} \)
79 \( 1 - 5.21T + 79T^{2} \)
83 \( 1 + 1.01T + 83T^{2} \)
89 \( 1 + 8.11T + 89T^{2} \)
97 \( 1 - 3.08T + 97T^{2} \)
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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

−8.120553243557477502301709261539, −7.58587307013528820528982438369, −6.51799774508259577206422790543, −6.02039106548312812999266859618, −5.03066859334790706599325262677, −3.95991335990975997654219200282, −3.54053654552260490617823223503, −2.69881150883645716203498345022, −1.32304337494271512809359859087, 0, 1.32304337494271512809359859087, 2.69881150883645716203498345022, 3.54053654552260490617823223503, 3.95991335990975997654219200282, 5.03066859334790706599325262677, 6.02039106548312812999266859618, 6.51799774508259577206422790543, 7.58587307013528820528982438369, 8.120553243557477502301709261539

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