Properties

Degree 2
Conductor $ 3 \cdot 7^{2} \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

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 + 2·13-s + 2·15-s − 16-s + 2·17-s + 18-s − 2·20-s + 22-s + 8·23-s − 3·24-s − 25-s + 2·26-s + 27-s − 6·29-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 + 0.554·13-s + 0.516·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.447·20-s + 0.213·22-s + 1.66·23-s − 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 1.11·29-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  =  0
Selberg data  =  $(2,\ 1617,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.112121701$
$L(\frac12)$  $\approx$  $3.112121701$
$L(\frac{3}{2})$   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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−19.90508728035994, −19.06991425617620, −18.56520302777783, −17.88707484329483, −17.22308613352113, −16.58791283017284, −15.54548954102634, −14.87766417647048, −14.43411512408130, −13.57331211702849, −13.34642346757835, −12.67955009753398, −11.78892670230951, −10.92749240423134, −9.861943929631486, −9.430206051937599, −8.732088684533911, −7.933480025037518, −6.787735198654673, −5.990451979645755, −5.259301363507105, −4.353384043516304, −3.442648140318228, −2.589439833982637, −1.200304482765721, 1.200304482765721, 2.589439833982637, 3.442648140318228, 4.353384043516304, 5.259301363507105, 5.990451979645755, 6.787735198654673, 7.933480025037518, 8.732088684533911, 9.430206051937599, 9.861943929631486, 10.92749240423134, 11.78892670230951, 12.67955009753398, 13.34642346757835, 13.57331211702849, 14.43411512408130, 14.87766417647048, 15.54548954102634, 16.58791283017284, 17.22308613352113, 17.88707484329483, 18.56520302777783, 19.06991425617620, 19.90508728035994

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