Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 11 \cdot 17^{2} $
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 − 4·7-s − 8-s + 9-s − 2·10-s + 11-s − 12-s + 4·14-s − 2·15-s + 16-s − 18-s + 6·19-s + 2·20-s + 4·21-s − 22-s + 24-s − 25-s − 27-s − 4·28-s + 6·29-s + 2·30-s + 2·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s − 0.288·12-s + 1.06·14-s − 0.516·15-s + 1/4·16-s − 0.235·18-s + 1.37·19-s + 0.447·20-s + 0.872·21-s − 0.213·22-s + 0.204·24-s − 1/5·25-s − 0.192·27-s − 0.755·28-s + 1.11·29-s + 0.365·30-s + 0.359·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19074 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19074 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(19074\)    =    \(2 \cdot 3 \cdot 11 \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{19074} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 19074,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.379917666\)
\(L(\frac12)\)  \(\approx\)  \(1.379917666\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
11 \( 1 - T \)
17 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−15.83909198612443, −15.67232309323681, −14.49127294264722, −14.16503447241227, −13.35092959662694, −13.02283908443415, −12.32989000005516, −11.90859304754840, −11.16265889613432, −10.64639367408742, −9.873990621611890, −9.583975218187676, −9.402206884473271, −8.477308827847364, −7.672461870805005, −7.082419353126970, −6.476093578096119, −5.944509825129585, −5.657226197038963, −4.616356826582130, −3.795676339197461, −2.896997799509848, −2.444334307098156, −1.244818611794759, −0.6360761419147270, 0.6360761419147270, 1.244818611794759, 2.444334307098156, 2.896997799509848, 3.795676339197461, 4.616356826582130, 5.657226197038963, 5.944509825129585, 6.476093578096119, 7.082419353126970, 7.672461870805005, 8.477308827847364, 9.402206884473271, 9.583975218187676, 9.873990621611890, 10.64639367408742, 11.16265889613432, 11.90859304754840, 12.32989000005516, 13.02283908443415, 13.35092959662694, 14.16503447241227, 14.49127294264722, 15.67232309323681, 15.83909198612443

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