Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 17 $
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 − 4·5-s + 6-s − 2·7-s − 8-s + 9-s + 4·10-s − 12-s − 6·13-s + 2·14-s + 4·15-s + 16-s − 17-s − 18-s + 4·19-s − 4·20-s + 2·21-s + 6·23-s + 24-s + 11·25-s + 6·26-s − 27-s − 2·28-s − 4·29-s − 4·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.288·12-s − 1.66·13-s + 0.534·14-s + 1.03·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.894·20-s + 0.436·21-s + 1.25·23-s + 0.204·24-s + 11/5·25-s + 1.17·26-s − 0.192·27-s − 0.377·28-s − 0.742·29-s − 0.730·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(102\)    =    \(2 \cdot 3 \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{102} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 102,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 + T \)
17 \( 1 + T \)
good5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−19.69554932097278, −19.20804948170250, −18.35187605411796, −17.02407738078465, −16.42521144271111, −15.51605117855991, −14.82213691937176, −12.85996017417381, −12.00111293806258, −11.31840597033056, −10.13636675520508, −8.951168622566808, −7.502241551585731, −6.996407801735695, −5.007168521649447, −3.345896165223093, 0, 3.345896165223093, 5.007168521649447, 6.996407801735695, 7.502241551585731, 8.951168622566808, 10.13636675520508, 11.31840597033056, 12.00111293806258, 12.85996017417381, 14.82213691937176, 15.51605117855991, 16.42521144271111, 17.02407738078465, 18.35187605411796, 19.20804948170250, 19.69554932097278

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