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  + 0.365·3-s + 1.83·5-s + 2.27·7-s − 2.86·9-s + 2.41·11-s − 4.42·13-s + 0.671·15-s − 4.27·17-s − 19-s + 0.830·21-s − 3.42·23-s − 1.62·25-s − 2.14·27-s − 5.90·29-s − 0.273·31-s + 0.880·33-s + 4.18·35-s − 5.93·37-s − 1.61·39-s − 0.691·41-s + 5.57·43-s − 5.26·45-s − 10.2·47-s − 1.82·49-s − 1.56·51-s + 53-s + 4.43·55-s + ⋯
L(s)  = 1  + 0.210·3-s + 0.822·5-s + 0.860·7-s − 0.955·9-s + 0.727·11-s − 1.22·13-s + 0.173·15-s − 1.03·17-s − 0.229·19-s + 0.181·21-s − 0.714·23-s − 0.324·25-s − 0.412·27-s − 1.09·29-s − 0.0491·31-s + 0.153·33-s + 0.707·35-s − 0.975·37-s − 0.258·39-s − 0.108·41-s + 0.849·43-s − 0.785·45-s − 1.49·47-s − 0.260·49-s − 0.218·51-s + 0.137·53-s + 0.598·55-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 \( 1 - 0.365T + 3T^{2} \)
5 \( 1 - 1.83T + 5T^{2} \)
7 \( 1 - 2.27T + 7T^{2} \)
11 \( 1 - 2.41T + 11T^{2} \)
13 \( 1 + 4.42T + 13T^{2} \)
17 \( 1 + 4.27T + 17T^{2} \)
23 \( 1 + 3.42T + 23T^{2} \)
29 \( 1 + 5.90T + 29T^{2} \)
31 \( 1 + 0.273T + 31T^{2} \)
37 \( 1 + 5.93T + 37T^{2} \)
41 \( 1 + 0.691T + 41T^{2} \)
43 \( 1 - 5.57T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
59 \( 1 + 4.01T + 59T^{2} \)
61 \( 1 + 10.2T + 61T^{2} \)
67 \( 1 - 2.97T + 67T^{2} \)
71 \( 1 + 0.559T + 71T^{2} \)
73 \( 1 + 8.72T + 73T^{2} \)
79 \( 1 - 0.654T + 79T^{2} \)
83 \( 1 - 3.92T + 83T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 - 1.00T + 97T^{2} \)
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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

−8.083403022090888919245264344236, −7.45217597385988565939209927953, −6.50395856333921446675409325559, −5.86815657823064844326982526971, −5.08260721690881972285150405173, −4.39454859909298790619525859807, −3.32991499040259663675187128222, −2.20287916561794040800393127710, −1.79541523958990391462715464688, 0, 1.79541523958990391462715464688, 2.20287916561794040800393127710, 3.32991499040259663675187128222, 4.39454859909298790619525859807, 5.08260721690881972285150405173, 5.86815657823064844326982526971, 6.50395856333921446675409325559, 7.45217597385988565939209927953, 8.083403022090888919245264344236

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