Properties

Degree 2
Conductor $ 3 \cdot 7 \cdot 11 \cdot 17^{2} $
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 − 7-s + 3·8-s + 9-s − 2·10-s + 11-s − 12-s + 6·13-s + 14-s + 2·15-s − 16-s − 18-s + 4·19-s − 2·20-s − 21-s − 22-s + 3·24-s − 25-s − 6·26-s + 27-s + 28-s + 2·29-s − 2·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.377·7-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 + 0.267·14-s + 0.516·15-s − 1/4·16-s − 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.218·21-s − 0.213·22-s + 0.612·24-s − 1/5·25-s − 1.17·26-s + 0.192·27-s + 0.188·28-s + 0.371·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 66759 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66759 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(66759\)    =    \(3 \cdot 7 \cdot 11 \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{66759} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 66759,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 - T \)
17 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
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

−14.36368454918740, −13.79297681024761, −13.45970310723112, −13.18900924154136, −12.70475310478959, −11.71247675870781, −11.49731204274751, −10.44360682211941, −10.37864948929318, −9.822587944615846, −9.134699008056965, −9.022382071499019, −8.410884623377210, −8.004627752648801, −7.198187783922560, −6.797015291415621, −6.100884990372716, −5.498463866378901, −5.043177798958870, −4.172066001924758, −3.511131922909687, −3.292884383624074, −2.117909700112148, −1.574332722806757, −1.091261639259896, 0, 1.091261639259896, 1.574332722806757, 2.117909700112148, 3.292884383624074, 3.511131922909687, 4.172066001924758, 5.043177798958870, 5.498463866378901, 6.100884990372716, 6.797015291415621, 7.198187783922560, 8.004627752648801, 8.410884623377210, 9.022382071499019, 9.134699008056965, 9.822587944615846, 10.37864948929318, 10.44360682211941, 11.49731204274751, 11.71247675870781, 12.70475310478959, 13.18900924154136, 13.45970310723112, 13.79297681024761, 14.36368454918740

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