Properties

Degree 2
Conductor $ 3 \cdot 7^{2} \cdot 11 $
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-s + 3-s − 4-s + 2·5-s − 6-s + 3·8-s + 9-s − 2·10-s − 11-s − 12-s − 6·13-s + 2·15-s − 16-s − 2·17-s − 18-s − 4·19-s − 2·20-s + 22-s + 3·24-s − 25-s + 6·26-s + 27-s − 2·29-s − 2·30-s − 8·31-s − 5·32-s − 33-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s − 0.288·12-s − 1.66·13-s + 0.516·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.213·22-s + 0.612·24-s − 1/5·25-s + 1.17·26-s + 0.192·27-s − 0.371·29-s − 0.365·30-s − 1.43·31-s − 0.883·32-s − 0.174·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1617} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 1617,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 \)
11 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−19.96128226323848, −19.46849171494552, −18.79623828222087, −18.12607613204859, −17.65319783842841, −16.81488408787753, −16.66603753979719, −15.30926277053960, −14.79098934378575, −14.10857196707475, −13.37419840231036, −12.95978126485134, −12.10658604034906, −10.91209706344318, −10.19431324320371, −9.681102187918904, −9.103413506990348, −8.384049684142721, −7.545091097063733, −6.887634145383234, −5.630516996249227, −4.851450033504389, −3.947614694751291, −2.508162950962352, −1.780100598714244, 0, 1.780100598714244, 2.508162950962352, 3.947614694751291, 4.851450033504389, 5.630516996249227, 6.887634145383234, 7.545091097063733, 8.384049684142721, 9.103413506990348, 9.681102187918904, 10.19431324320371, 10.91209706344318, 12.10658604034906, 12.95978126485134, 13.37419840231036, 14.10857196707475, 14.79098934378575, 15.30926277053960, 16.66603753979719, 16.81488408787753, 17.65319783842841, 18.12607613204859, 18.79623828222087, 19.46849171494552, 19.96128226323848

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