Properties

Degree 2
Conductor $ 3 \cdot 11^{2} \cdot 13^{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 − 3-s − 4-s + 6-s + 3·8-s + 9-s + 12-s − 16-s + 4·17-s − 18-s − 8·19-s − 3·24-s − 5·25-s − 27-s − 4·29-s + 6·31-s − 5·32-s − 4·34-s − 36-s + 6·37-s + 8·38-s + 6·41-s + 2·43-s + 8·47-s + 48-s − 7·49-s + 5·50-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.288·12-s − 1/4·16-s + 0.970·17-s − 0.235·18-s − 1.83·19-s − 0.612·24-s − 25-s − 0.192·27-s − 0.742·29-s + 1.07·31-s − 0.883·32-s − 0.685·34-s − 1/6·36-s + 0.986·37-s + 1.29·38-s + 0.937·41-s + 0.304·43-s + 1.16·47-s + 0.144·48-s − 49-s + 0.707·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(61347\)    =    \(3 \cdot 11^{2} \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{61347} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 61347,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.8561973530\)
\(L(\frac12)\)  \(\approx\)  \(0.8561973530\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;11,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
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

−14.26414746510577, −13.78285329581294, −13.19089962633151, −12.65963020379568, −12.46139304007939, −11.47050805537366, −11.32759072945404, −10.54742255677710, −10.15819475080112, −9.783398060356535, −9.188935944265149, −8.583196523419426, −8.182145044528217, −7.532356243470956, −7.216196239385768, −6.194080314128139, −6.027988343766937, −5.262469621032710, −4.619541782544398, −4.108473355204198, −3.655973211307016, −2.539280480060932, −1.928679951309421, −1.056649571768570, −0.4420975888599965, 0.4420975888599965, 1.056649571768570, 1.928679951309421, 2.539280480060932, 3.655973211307016, 4.108473355204198, 4.619541782544398, 5.262469621032710, 6.027988343766937, 6.194080314128139, 7.216196239385768, 7.532356243470956, 8.182145044528217, 8.583196523419426, 9.188935944265149, 9.783398060356535, 10.15819475080112, 10.54742255677710, 11.32759072945404, 11.47050805537366, 12.46139304007939, 12.65963020379568, 13.19089962633151, 13.78285329581294, 14.26414746510577

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