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  − 3-s − 3·5-s + 7-s − 2·9-s + 2·11-s − 2·13-s + 3·15-s − 17-s + 19-s − 21-s + 8·23-s + 4·25-s + 5·27-s − 29-s − 2·33-s − 3·35-s − 2·37-s + 2·39-s − 7·41-s + 8·43-s + 6·45-s + 8·47-s − 6·49-s + 51-s + 3·53-s − 6·55-s − 57-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s + 0.377·7-s − 2/3·9-s + 0.603·11-s − 0.554·13-s + 0.774·15-s − 0.242·17-s + 0.229·19-s − 0.218·21-s + 1.66·23-s + 4/5·25-s + 0.962·27-s − 0.185·29-s − 0.348·33-s − 0.507·35-s − 0.328·37-s + 0.320·39-s − 1.09·41-s + 1.21·43-s + 0.894·45-s + 1.16·47-s − 6/7·49-s + 0.140·51-s + 0.412·53-s − 0.809·55-s − 0.132·57-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 \( 1 + T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−8.095575659078958781344328488228, −7.23427701735385486286027819341, −6.82930689171494415309635766595, −5.75281845521053839460736579453, −5.02872490792614591051309557729, −4.34515175995065085137023566002, −3.49127357774111825372602476656, −2.62377678048319806853132040337, −1.11782411776987817886590080677, 0, 1.11782411776987817886590080677, 2.62377678048319806853132040337, 3.49127357774111825372602476656, 4.34515175995065085137023566002, 5.02872490792614591051309557729, 5.75281845521053839460736579453, 6.82930689171494415309635766595, 7.23427701735385486286027819341, 8.095575659078958781344328488228

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