Properties

Degree 2
Conductor $ 2 \cdot 7 \cdot 29^{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 + 2·3-s + 4-s + 2·6-s + 7-s + 8-s + 9-s + 2·12-s − 4·13-s + 14-s + 16-s − 6·17-s + 18-s − 2·19-s + 2·21-s + 2·24-s − 5·25-s − 4·26-s − 4·27-s + 28-s + 4·31-s + 32-s − 6·34-s + 36-s − 2·37-s − 2·38-s − 8·39-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.577·12-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.458·19-s + 0.436·21-s + 0.408·24-s − 25-s − 0.784·26-s − 0.769·27-s + 0.188·28-s + 0.718·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s − 0.324·38-s − 1.28·39-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 11774 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 11774 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(11774\)    =    \(2 \cdot 7 \cdot 29^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{11774} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 11774,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;29\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 - T \)
29 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−16.77647427893458, −15.64556738698007, −15.41002470155411, −14.98711878204920, −14.35349627079082, −13.86725723863853, −13.42210000136659, −12.95748522632581, −12.00219832830320, −11.80830326510315, −10.96206395528080, −10.30235038181251, −9.710123309500313, −8.874952487363077, −8.574673996165233, −7.738998868747291, −7.301180852955309, −6.553154135575785, −5.821693502833276, −4.956662826024917, −4.395328602068130, −3.753996760736394, −2.866516421254112, −2.311589921492336, −1.715204312746312, 0, 1.715204312746312, 2.311589921492336, 2.866516421254112, 3.753996760736394, 4.395328602068130, 4.956662826024917, 5.821693502833276, 6.553154135575785, 7.301180852955309, 7.738998868747291, 8.574673996165233, 8.874952487363077, 9.710123309500313, 10.30235038181251, 10.96206395528080, 11.80830326510315, 12.00219832830320, 12.95748522632581, 13.42210000136659, 13.86725723863853, 14.35349627079082, 14.98711878204920, 15.41002470155411, 15.64556738698007, 16.77647427893458

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