Properties

Degree 2
Conductor $ 2 \cdot 7^{2} \cdot 11^{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 + 2·3-s + 4-s + 2·6-s + 8-s + 9-s + 2·12-s − 4·13-s + 16-s + 6·17-s + 18-s + 2·19-s + 2·24-s − 5·25-s − 4·26-s − 4·27-s + 6·29-s + 4·31-s + 32-s + 6·34-s + 36-s + 2·37-s + 2·38-s − 8·39-s + 6·41-s − 8·43-s + 12·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s + 1/3·9-s + 0.577·12-s − 1.10·13-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.458·19-s + 0.408·24-s − 25-s − 0.784·26-s − 0.769·27-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 − 1.28·39-s + 0.937·41-s − 1.21·43-s + 1.75·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11858 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11858 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(11858\)    =    \(2 \cdot 7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{11858} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 11858,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.419076915$
$L(\frac12)$  $\approx$  $5.419076915$
$L(\frac{3}{2})$   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 \( 1 \)
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

−16.05913833459706, −15.80525404670970, −14.90092823984905, −14.70471688429025, −14.15525272390521, −13.66293701257632, −13.21793037602701, −12.30127686725597, −12.04147499416210, −11.45995004372553, −10.48185724323955, −9.841312076117640, −9.609502670858404, −8.590184015029907, −8.138332901724243, −7.445136172819586, −7.098223422001412, −5.991946046851054, −5.521752848931414, −4.705055595464382, −3.978185445514306, −3.269592895833760, −2.682325440770486, −2.074439482312945, −0.9015889709166229, 0.9015889709166229, 2.074439482312945, 2.682325440770486, 3.269592895833760, 3.978185445514306, 4.705055595464382, 5.521752848931414, 5.991946046851054, 7.098223422001412, 7.445136172819586, 8.138332901724243, 8.590184015029907, 9.609502670858404, 9.841312076117640, 10.48185724323955, 11.45995004372553, 12.04147499416210, 12.30127686725597, 13.21793037602701, 13.66293701257632, 14.15525272390521, 14.70471688429025, 14.90092823984905, 15.80525404670970, 16.05913833459706

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