Properties

Degree 2
Conductor $ 2 \cdot 7 \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 − 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 + 6·29-s − 4·31-s + 32-s − 6·34-s + 36-s + 2·37-s − 2·38-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 + 1.11·29-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 + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1694 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1694\)    =    \(2 \cdot 7 \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1694} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 1694,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
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

−19.94367884576932, −19.69442509071907, −18.56770425748017, −17.98234394203928, −17.42409444281153, −16.60105809299969, −16.14857892256002, −15.48322661353151, −14.86949690416681, −13.73312072265061, −13.38555188081941, −12.62364411152437, −11.84417045300307, −11.31241460864229, −10.74827747141931, −10.01782448311884, −8.888226006934559, −8.145906800879012, −6.823979641914685, −6.424553312164633, −5.763213506174021, −4.859907514669799, −4.090523656201600, −3.030397617533132, −1.671205539809654, 0, 1.671205539809654, 3.030397617533132, 4.090523656201600, 4.859907514669799, 5.763213506174021, 6.424553312164633, 6.823979641914685, 8.145906800879012, 8.888226006934559, 10.01782448311884, 10.74827747141931, 11.31241460864229, 11.84417045300307, 12.62364411152437, 13.38555188081941, 13.73312072265061, 14.86949690416681, 15.48322661353151, 16.14857892256002, 16.60105809299969, 17.42409444281153, 17.98234394203928, 18.56770425748017, 19.69442509071907, 19.94367884576932

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