Properties

Degree 2
Conductor 17
Sign $-1$
Motivic weight 3
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 8·3-s + 4-s + 6·5-s + 24·6-s − 28·7-s + 21·8-s + 37·9-s − 18·10-s − 24·11-s − 8·12-s − 58·13-s + 84·14-s − 48·15-s − 71·16-s + 17·17-s − 111·18-s + 116·19-s + 6·20-s + 224·21-s + 72·22-s − 60·23-s − 168·24-s − 89·25-s + 174·26-s − 80·27-s − 28·28-s + ⋯
L(s)  = 1  − 1.06·2-s − 1.53·3-s + 1/8·4-s + 0.536·5-s + 1.63·6-s − 1.51·7-s + 0.928·8-s + 1.37·9-s − 0.569·10-s − 0.657·11-s − 0.192·12-s − 1.23·13-s + 1.60·14-s − 0.826·15-s − 1.10·16-s + 0.242·17-s − 1.45·18-s + 1.40·19-s + 0.0670·20-s + 2.32·21-s + 0.697·22-s − 0.543·23-s − 1.42·24-s − 0.711·25-s + 1.31·26-s − 0.570·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(3\)
character  :  $\chi_{17} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 17,\ (\ :3/2),\ -1)$
$L(2)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{5}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 17$, \(F_p\) is a polynomial of degree 2. If $p = 17$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad17 \( 1 - p T \)
good2 \( 1 + 3 T + p^{3} T^{2} \)
3 \( 1 + 8 T + p^{3} T^{2} \)
5 \( 1 - 6 T + p^{3} T^{2} \)
7 \( 1 + 4 p T + p^{3} T^{2} \)
11 \( 1 + 24 T + p^{3} T^{2} \)
13 \( 1 + 58 T + p^{3} T^{2} \)
19 \( 1 - 116 T + p^{3} T^{2} \)
23 \( 1 + 60 T + p^{3} T^{2} \)
29 \( 1 - 30 T + p^{3} T^{2} \)
31 \( 1 + 172 T + p^{3} T^{2} \)
37 \( 1 + 58 T + p^{3} T^{2} \)
41 \( 1 + 342 T + p^{3} T^{2} \)
43 \( 1 + 148 T + p^{3} T^{2} \)
47 \( 1 - 288 T + p^{3} T^{2} \)
53 \( 1 - 6 p T + p^{3} T^{2} \)
59 \( 1 - 252 T + p^{3} T^{2} \)
61 \( 1 - 110 T + p^{3} T^{2} \)
67 \( 1 + 484 T + p^{3} T^{2} \)
71 \( 1 + 708 T + p^{3} T^{2} \)
73 \( 1 - 362 T + p^{3} T^{2} \)
79 \( 1 + 484 T + p^{3} T^{2} \)
83 \( 1 - 756 T + p^{3} T^{2} \)
89 \( 1 + 774 T + p^{3} T^{2} \)
97 \( 1 + 382 T + p^{3} T^{2} \)
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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

−17.74385007257172020163854715717, −16.76846225019868675393598056767, −16.01469057900752012644493569831, −13.41820326822456031422724556873, −12.07668098573410522658047714136, −10.28975788767548631464421204802, −9.658330125586286234533976808491, −7.15720926687727428265062529690, −5.46822905834790594613537249224, 0, 5.46822905834790594613537249224, 7.15720926687727428265062529690, 9.658330125586286234533976808491, 10.28975788767548631464421204802, 12.07668098573410522658047714136, 13.41820326822456031422724556873, 16.01469057900752012644493569831, 16.76846225019868675393598056767, 17.74385007257172020163854715717

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