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  − 1.01·3-s + 3.39·5-s − 2.78·7-s − 1.96·9-s + 1.42·11-s − 5.82·13-s − 3.46·15-s + 0.730·17-s − 19-s + 2.83·21-s + 2.62·23-s + 6.53·25-s + 5.05·27-s + 9.98·29-s + 8.88·31-s − 1.45·33-s − 9.44·35-s − 1.61·37-s + 5.93·39-s + 0.419·41-s − 9.18·43-s − 6.66·45-s − 7.19·47-s + 0.735·49-s − 0.743·51-s + 53-s + 4.84·55-s + ⋯
L(s)  = 1  − 0.588·3-s + 1.51·5-s − 1.05·7-s − 0.653·9-s + 0.430·11-s − 1.61·13-s − 0.893·15-s + 0.177·17-s − 0.229·19-s + 0.618·21-s + 0.547·23-s + 1.30·25-s + 0.973·27-s + 1.85·29-s + 1.59·31-s − 0.253·33-s − 1.59·35-s − 0.265·37-s + 0.950·39-s + 0.0655·41-s − 1.40·43-s − 0.992·45-s − 1.05·47-s + 0.105·49-s − 0.104·51-s + 0.137·53-s + 0.653·55-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 \( 1 + 1.01T + 3T^{2} \)
5 \( 1 - 3.39T + 5T^{2} \)
7 \( 1 + 2.78T + 7T^{2} \)
11 \( 1 - 1.42T + 11T^{2} \)
13 \( 1 + 5.82T + 13T^{2} \)
17 \( 1 - 0.730T + 17T^{2} \)
23 \( 1 - 2.62T + 23T^{2} \)
29 \( 1 - 9.98T + 29T^{2} \)
31 \( 1 - 8.88T + 31T^{2} \)
37 \( 1 + 1.61T + 37T^{2} \)
41 \( 1 - 0.419T + 41T^{2} \)
43 \( 1 + 9.18T + 43T^{2} \)
47 \( 1 + 7.19T + 47T^{2} \)
59 \( 1 + 11.5T + 59T^{2} \)
61 \( 1 - 0.616T + 61T^{2} \)
67 \( 1 + 12.1T + 67T^{2} \)
71 \( 1 + 8.09T + 71T^{2} \)
73 \( 1 - 1.60T + 73T^{2} \)
79 \( 1 + 3.28T + 79T^{2} \)
83 \( 1 + 13.0T + 83T^{2} \)
89 \( 1 - 1.73T + 89T^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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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

−8.189885536618771440876882922435, −6.95814901821403471490013301429, −6.46654138701629162146964653749, −6.02056788119277494325569426416, −5.12080889468903488925010492132, −4.62420562924286314352938644873, −2.96787168060354296635833236344, −2.70283699403369309281007441944, −1.38153870052590982146588171181, 0, 1.38153870052590982146588171181, 2.70283699403369309281007441944, 2.96787168060354296635833236344, 4.62420562924286314352938644873, 5.12080889468903488925010492132, 6.02056788119277494325569426416, 6.46654138701629162146964653749, 6.95814901821403471490013301429, 8.189885536618771440876882922435

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