Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 7^{2} \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  + 2·5-s − 2·13-s − 4·17-s − 6·19-s − 25-s − 8·29-s + 8·31-s + 10·37-s − 8·41-s + 2·43-s − 8·47-s + 2·53-s + 12·59-s + 10·61-s − 4·65-s + 12·67-s − 8·71-s + 6·73-s + 2·79-s − 16·83-s − 8·85-s − 14·89-s − 12·95-s + 2·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.554·13-s − 0.970·17-s − 1.37·19-s − 1/5·25-s − 1.48·29-s + 1.43·31-s + 1.64·37-s − 1.24·41-s + 0.304·43-s − 1.16·47-s + 0.274·53-s + 1.56·59-s + 1.28·61-s − 0.496·65-s + 1.46·67-s − 0.949·71-s + 0.702·73-s + 0.225·79-s − 1.75·83-s − 0.867·85-s − 1.48·89-s − 1.23·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(213444\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{213444} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 213444,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.24798290167523, −12.89587202006054, −12.47801802239768, −11.65949540630467, −11.37485696331492, −11.02114540976496, −10.17735601989125, −10.01160452457469, −9.616369665629518, −9.004794520440899, −8.508206680736385, −8.150791895625306, −7.526379680576695, −6.776278633878490, −6.628934478844487, −6.031187091820926, −5.505522394297602, −5.052934717318876, −4.280964496134155, −4.131254581599741, −3.246495922274279, −2.511152660347963, −2.168794766565716, −1.703658572803272, −0.7633386040796912, 0, 0.7633386040796912, 1.703658572803272, 2.168794766565716, 2.511152660347963, 3.246495922274279, 4.131254581599741, 4.280964496134155, 5.052934717318876, 5.505522394297602, 6.031187091820926, 6.628934478844487, 6.776278633878490, 7.526379680576695, 8.150791895625306, 8.508206680736385, 9.004794520440899, 9.616369665629518, 10.01160452457469, 10.17735601989125, 11.02114540976496, 11.37485696331492, 11.65949540630467, 12.47801802239768, 12.89587202006054, 13.24798290167523

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