Properties

Degree 2
Conductor $ 3 \cdot 5 $
Sign $-1$
Motivic weight 9
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 81·3-s − 496·4-s + 625·5-s − 324·6-s − 7.68e3·7-s + 4.03e3·8-s + 6.56e3·9-s − 2.50e3·10-s − 8.64e4·11-s − 4.01e4·12-s − 1.49e5·13-s + 3.07e4·14-s + 5.06e4·15-s + 2.37e5·16-s − 2.07e5·17-s − 2.62e4·18-s + 7.16e5·19-s − 3.10e5·20-s − 6.22e5·21-s + 3.45e5·22-s + 1.36e6·23-s + 3.26e5·24-s + 3.90e5·25-s + 5.99e5·26-s + 5.31e5·27-s + 3.80e6·28-s + ⋯
L(s)  = 1  − 0.176·2-s + 0.577·3-s − 0.968·4-s + 0.447·5-s − 0.102·6-s − 1.20·7-s + 0.348·8-s + 1/3·9-s − 0.0790·10-s − 1.77·11-s − 0.559·12-s − 1.45·13-s + 0.213·14-s + 0.258·15-s + 0.907·16-s − 0.602·17-s − 0.0589·18-s + 1.26·19-s − 0.433·20-s − 0.698·21-s + 0.314·22-s + 1.02·23-s + 0.200·24-s + 1/5·25-s + 0.257·26-s + 0.192·27-s + 1.17·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(15\)    =    \(3 \cdot 5\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(9\)
character  :  $\chi_{15} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 15,\ (\ :9/2),\ -1)\)
\(L(5)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{11}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - p^{4} T \)
5 \( 1 - p^{4} T \)
good2 \( 1 + p^{2} T + p^{9} T^{2} \)
7 \( 1 + 7680 T + p^{9} T^{2} \)
11 \( 1 + 86404 T + p^{9} T^{2} \)
13 \( 1 + 149978 T + p^{9} T^{2} \)
17 \( 1 + 207622 T + p^{9} T^{2} \)
19 \( 1 - 716284 T + p^{9} T^{2} \)
23 \( 1 - 1369920 T + p^{9} T^{2} \)
29 \( 1 + 3194402 T + p^{9} T^{2} \)
31 \( 1 + 2349000 T + p^{9} T^{2} \)
37 \( 1 - 18735710 T + p^{9} T^{2} \)
41 \( 1 + 29282630 T + p^{9} T^{2} \)
43 \( 1 + 1516724 T + p^{9} T^{2} \)
47 \( 1 - 615752 T + p^{9} T^{2} \)
53 \( 1 - 4747430 T + p^{9} T^{2} \)
59 \( 1 - 60616076 T + p^{9} T^{2} \)
61 \( 1 + 126745682 T + p^{9} T^{2} \)
67 \( 1 + 111182652 T + p^{9} T^{2} \)
71 \( 1 + 175551608 T + p^{9} T^{2} \)
73 \( 1 + 61233350 T + p^{9} T^{2} \)
79 \( 1 - 234431160 T + p^{9} T^{2} \)
83 \( 1 - 118910388 T + p^{9} T^{2} \)
89 \( 1 + 316534326 T + p^{9} T^{2} \)
97 \( 1 - 242912258 T + p^{9} T^{2} \)
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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

−16.56626842229995994392970499777, −15.06452120901311457156989885385, −13.47945500338065112280866788268, −12.85360330875681719091464429979, −10.13027685362140146444702249976, −9.292423620461158919117509450076, −7.52833366745717521206717960305, −5.13826459498362935951837990090, −2.89046851552116422920781524476, 0, 2.89046851552116422920781524476, 5.13826459498362935951837990090, 7.52833366745717521206717960305, 9.292423620461158919117509450076, 10.13027685362140146444702249976, 12.85360330875681719091464429979, 13.47945500338065112280866788268, 15.06452120901311457156989885385, 16.56626842229995994392970499777

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