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  + 0.0800·3-s + 0.442·5-s + 2.60·7-s − 2.99·9-s − 2.05·11-s + 2.33·13-s + 0.0354·15-s + 2.04·17-s − 19-s + 0.208·21-s − 6.75·23-s − 4.80·25-s − 0.479·27-s − 0.161·29-s − 7.92·31-s − 0.164·33-s + 1.15·35-s + 3.74·37-s + 0.187·39-s − 5.17·41-s − 4.07·43-s − 1.32·45-s + 2.84·47-s − 0.213·49-s + 0.163·51-s + 53-s − 0.909·55-s + ⋯
L(s)  = 1  + 0.0462·3-s + 0.198·5-s + 0.984·7-s − 0.997·9-s − 0.619·11-s + 0.648·13-s + 0.00915·15-s + 0.494·17-s − 0.229·19-s + 0.0455·21-s − 1.40·23-s − 0.960·25-s − 0.0923·27-s − 0.0299·29-s − 1.42·31-s − 0.0286·33-s + 0.194·35-s + 0.615·37-s + 0.0299·39-s − 0.807·41-s − 0.621·43-s − 0.197·45-s + 0.415·47-s − 0.0304·49-s + 0.0228·51-s + 0.137·53-s − 0.122·55-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 \( 1 - 0.0800T + 3T^{2} \)
5 \( 1 - 0.442T + 5T^{2} \)
7 \( 1 - 2.60T + 7T^{2} \)
11 \( 1 + 2.05T + 11T^{2} \)
13 \( 1 - 2.33T + 13T^{2} \)
17 \( 1 - 2.04T + 17T^{2} \)
23 \( 1 + 6.75T + 23T^{2} \)
29 \( 1 + 0.161T + 29T^{2} \)
31 \( 1 + 7.92T + 31T^{2} \)
37 \( 1 - 3.74T + 37T^{2} \)
41 \( 1 + 5.17T + 41T^{2} \)
43 \( 1 + 4.07T + 43T^{2} \)
47 \( 1 - 2.84T + 47T^{2} \)
59 \( 1 - 8.26T + 59T^{2} \)
61 \( 1 + 4.20T + 61T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 + 4.25T + 71T^{2} \)
73 \( 1 - 2.99T + 73T^{2} \)
79 \( 1 - 1.77T + 79T^{2} \)
83 \( 1 - 13.1T + 83T^{2} \)
89 \( 1 + 8.31T + 89T^{2} \)
97 \( 1 + 13.1T + 97T^{2} \)
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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

−8.031675795909423293537657615703, −7.65910858744341885570813218049, −6.48369071033616105758739100950, −5.67712230865080370706950935751, −5.31525065232321712725580273921, −4.23018258424691325986437697642, −3.43296991264731113104034989191, −2.35774962156931810740124080077, −1.57461544171146835514338887297, 0, 1.57461544171146835514338887297, 2.35774962156931810740124080077, 3.43296991264731113104034989191, 4.23018258424691325986437697642, 5.31525065232321712725580273921, 5.67712230865080370706950935751, 6.48369071033616105758739100950, 7.65910858744341885570813218049, 8.031675795909423293537657615703

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