Properties

Degree 2
Conductor $ 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  + 2·2-s + 2·4-s − 2·5-s − 2·7-s − 3·9-s − 4·10-s − 3·11-s + 4·13-s − 4·14-s − 4·16-s + 17-s − 6·18-s − 19-s − 4·20-s − 6·22-s + 23-s − 25-s + 8·26-s − 4·28-s − 6·29-s − 8·31-s − 8·32-s + 2·34-s + 4·35-s − 6·36-s − 2·37-s − 2·38-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.894·5-s − 0.755·7-s − 9-s − 1.26·10-s − 0.904·11-s + 1.10·13-s − 1.06·14-s − 16-s + 0.242·17-s − 1.41·18-s − 0.229·19-s − 0.894·20-s − 1.27·22-s + 0.208·23-s − 1/5·25-s + 1.56·26-s − 0.755·28-s − 1.11·29-s − 1.43·31-s − 1.41·32-s + 0.342·34-s + 0.676·35-s − 36-s − 0.328·37-s − 0.324·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1003\)    =    \(17 \cdot 59\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1003} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1003,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{17,\;59\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{17,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad17 \( 1 - T \)
59 \( 1 - T \)
good2 \( 1 - p T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 15 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 19 T + p T^{2} \)
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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

−9.491089311396895428526430046218, −8.643308621193790780325039006644, −7.76127111271616858613177130690, −6.76143361930033746876890443837, −5.78395595155708678947454558582, −5.32342147634490412558056408515, −3.93529221306480405675404880745, −3.51670357488875386563464505592, −2.50500987736200068031649270207, 0, 2.50500987736200068031649270207, 3.51670357488875386563464505592, 3.93529221306480405675404880745, 5.32342147634490412558056408515, 5.78395595155708678947454558582, 6.76143361930033746876890443837, 7.76127111271616858613177130690, 8.643308621193790780325039006644, 9.491089311396895428526430046218

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