Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.859·3-s + 2.49·5-s − 3.53·7-s − 2.26·9-s + 1.54·11-s + 3.17·13-s − 2.14·15-s − 17-s + 0.523·19-s + 3.03·21-s − 5.23·23-s + 1.24·25-s + 4.52·27-s + 4.96·29-s + 3.14·31-s − 1.32·33-s − 8.82·35-s − 7.80·37-s − 2.72·39-s − 1.58·41-s − 1.96·43-s − 5.64·45-s + 2.57·47-s + 5.48·49-s + 0.859·51-s + 2.58·53-s + 3.85·55-s + ⋯
L(s)  = 1  − 0.496·3-s + 1.11·5-s − 1.33·7-s − 0.753·9-s + 0.464·11-s + 0.880·13-s − 0.554·15-s − 0.242·17-s + 0.120·19-s + 0.663·21-s − 1.09·23-s + 0.248·25-s + 0.870·27-s + 0.921·29-s + 0.564·31-s − 0.230·33-s − 1.49·35-s − 1.28·37-s − 0.436·39-s − 0.248·41-s − 0.299·43-s − 0.841·45-s + 0.376·47-s + 0.784·49-s + 0.120·51-s + 0.355·53-s + 0.519·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;17,\;59\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;17,\;59\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
17 \( 1 + T \)
59 \( 1 - T \)
good3 \( 1 + 0.859T + 3T^{2} \)
5 \( 1 - 2.49T + 5T^{2} \)
7 \( 1 + 3.53T + 7T^{2} \)
11 \( 1 - 1.54T + 11T^{2} \)
13 \( 1 - 3.17T + 13T^{2} \)
19 \( 1 - 0.523T + 19T^{2} \)
23 \( 1 + 5.23T + 23T^{2} \)
29 \( 1 - 4.96T + 29T^{2} \)
31 \( 1 - 3.14T + 31T^{2} \)
37 \( 1 + 7.80T + 37T^{2} \)
41 \( 1 + 1.58T + 41T^{2} \)
43 \( 1 + 1.96T + 43T^{2} \)
47 \( 1 - 2.57T + 47T^{2} \)
53 \( 1 - 2.58T + 53T^{2} \)
61 \( 1 - 6.54T + 61T^{2} \)
67 \( 1 + 5.68T + 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 + 1.37T + 73T^{2} \)
79 \( 1 + 6.64T + 79T^{2} \)
83 \( 1 + 16.8T + 83T^{2} \)
89 \( 1 - 2.73T + 89T^{2} \)
97 \( 1 - 3.59T + 97T^{2} \)
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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

−8.331559340468110095993848868804, −6.99428823283412175178905440786, −6.41576472722736345329033419043, −5.95227349140632384642525898146, −5.42386655299874937769558624630, −4.24027292176833046285234201450, −3.31331383234736153623435954202, −2.53526196608386433408047714065, −1.37245905431549897903115807811, 0, 1.37245905431549897903115807811, 2.53526196608386433408047714065, 3.31331383234736153623435954202, 4.24027292176833046285234201450, 5.42386655299874937769558624630, 5.95227349140632384642525898146, 6.41576472722736345329033419043, 6.99428823283412175178905440786, 8.331559340468110095993848868804

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