Properties

Degree 2
Conductor $ 2 \cdot 11^{2} \cdot 17 $
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 + 4·7-s − 8-s + 9-s − 2·12-s − 2·13-s − 4·14-s + 16-s + 17-s − 18-s + 4·19-s − 8·21-s + 2·24-s − 5·25-s + 2·26-s + 4·27-s + 4·28-s − 4·31-s − 32-s − 34-s + 36-s − 4·37-s − 4·38-s + 4·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.577·12-s − 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s − 1.74·21-s + 0.408·24-s − 25-s + 0.392·26-s + 0.769·27-s + 0.755·28-s − 0.718·31-s − 0.176·32-s − 0.171·34-s + 1/6·36-s − 0.657·37-s − 0.648·38-s + 0.640·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4114\)    =    \(2 \cdot 11^{2} \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4114} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 4114,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;11,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;11,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
11 \( 1 \)
17 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−18.20345945485661, −17.67080714141629, −17.26225459257094, −16.81964826900180, −16.09740893867739, −15.48368077699114, −14.70787030496930, −14.22309555933766, −13.43231584484641, −12.34463896506211, −11.91206362737648, −11.37135892832404, −10.99535063675664, −10.17490230410295, −9.641335387313975, −8.624363935884874, −8.076455834435978, −7.364974837247333, −6.720844523627733, −5.673207945109312, −5.268949928538387, −4.561978362663805, −3.346116681674289, −2.050229112924842, −1.252847766550833, 0, 1.252847766550833, 2.050229112924842, 3.346116681674289, 4.561978362663805, 5.268949928538387, 5.673207945109312, 6.720844523627733, 7.364974837247333, 8.076455834435978, 8.624363935884874, 9.641335387313975, 10.17490230410295, 10.99535063675664, 11.37135892832404, 11.91206362737648, 12.34463896506211, 13.43231584484641, 14.22309555933766, 14.70787030496930, 15.48368077699114, 16.09740893867739, 16.81964826900180, 17.26225459257094, 17.67080714141629, 18.20345945485661

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