Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11^{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 − 5-s + 6-s − 8-s + 9-s + 10-s − 12-s − 2·13-s + 15-s + 16-s + 2·17-s − 18-s + 4·19-s − 20-s − 8·23-s + 24-s + 25-s + 2·26-s − 27-s + 2·29-s − 30-s − 32-s − 2·34-s + 36-s + 6·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.554·13-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.371·29-s − 0.182·30-s − 0.176·32-s − 0.342·34-s + 1/6·36-s + 0.986·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(177870\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{177870} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 177870,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 \)
good13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−13.40872903179907, −12.65870745906686, −12.36201119943410, −11.84300169340140, −11.61641719388904, −11.03583487946795, −10.58110870971255, −9.881539747439572, −9.829850584996612, −9.286610117916261, −8.445487381624618, −8.206487169077241, −7.567617883681788, −7.290852378562945, −6.737809564860694, −5.987671801911612, −5.814572840099754, −5.095934268670968, −4.459088241978366, −4.062770496523932, −3.189364062085641, −2.860590431625262, −1.945700521908398, −1.431976275845841, −0.6499595355473003, 0, 0.6499595355473003, 1.431976275845841, 1.945700521908398, 2.860590431625262, 3.189364062085641, 4.062770496523932, 4.459088241978366, 5.095934268670968, 5.814572840099754, 5.987671801911612, 6.737809564860694, 7.290852378562945, 7.567617883681788, 8.206487169077241, 8.445487381624618, 9.286610117916261, 9.829850584996612, 9.881539747439572, 10.58110870971255, 11.03583487946795, 11.61641719388904, 11.84300169340140, 12.36201119943410, 12.65870745906686, 13.40872903179907

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