Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.369 + 0.929i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.0974·3-s − 3.46i·5-s − 2.64i·7-s − 8.99·9-s + 2.92·11-s − 19.1i·13-s + 0.337i·15-s − 14.3·17-s − 8.09·19-s + 0.257i·21-s − 16.7i·23-s + 12.9·25-s + 1.75·27-s − 27.1i·29-s − 44.8i·31-s + ⋯
L(s)  = 1  − 0.0324·3-s − 0.693i·5-s − 0.377i·7-s − 0.998·9-s + 0.266·11-s − 1.47i·13-s + 0.0225i·15-s − 0.846·17-s − 0.426·19-s + 0.0122i·21-s − 0.728i·23-s + 0.519·25-s + 0.0649·27-s − 0.936i·29-s − 1.44i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.369 + 0.929i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.369 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.369 + 0.929i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (15, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.369 + 0.929i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.602716 - 0.888399i\)
\(L(\frac12)\)  \(\approx\)  \(0.602716 - 0.888399i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + 2.64iT \)
good3 \( 1 + 0.0974T + 9T^{2} \)
5 \( 1 + 3.46iT - 25T^{2} \)
11 \( 1 - 2.92T + 121T^{2} \)
13 \( 1 + 19.1iT - 169T^{2} \)
17 \( 1 + 14.3T + 289T^{2} \)
19 \( 1 + 8.09T + 361T^{2} \)
23 \( 1 + 16.7iT - 529T^{2} \)
29 \( 1 + 27.1iT - 841T^{2} \)
31 \( 1 + 44.8iT - 961T^{2} \)
37 \( 1 - 39.5iT - 1.36e3T^{2} \)
41 \( 1 - 45.8T + 1.68e3T^{2} \)
43 \( 1 + 61.0T + 1.84e3T^{2} \)
47 \( 1 - 46.2iT - 2.20e3T^{2} \)
53 \( 1 - 9.69iT - 2.80e3T^{2} \)
59 \( 1 - 114.T + 3.48e3T^{2} \)
61 \( 1 + 7.48iT - 3.72e3T^{2} \)
67 \( 1 - 12.0T + 4.48e3T^{2} \)
71 \( 1 - 129. iT - 5.04e3T^{2} \)
73 \( 1 + 18.2T + 5.32e3T^{2} \)
79 \( 1 - 42.6iT - 6.24e3T^{2} \)
83 \( 1 - 109.T + 6.88e3T^{2} \)
89 \( 1 + 80.9T + 7.92e3T^{2} \)
97 \( 1 - 162.T + 9.40e3T^{2} \)
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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

−11.68144893850633224132608908252, −10.84008908556587156733379040591, −9.776473367082583590343679032462, −8.607263933313369070125916140874, −7.995001070140867622430196691524, −6.49029251477833633801649066198, −5.43827208392922188699097702217, −4.25416233653326295398420032189, −2.68833386320037761770043497563, −0.57262552769252382739705012423, 2.14560546054751703566061367445, 3.52368282682146355357071823154, 5.02333227321736714061387694455, 6.35116874273869545076090233449, 7.07104014206124897483691832442, 8.627857133823402794699781261214, 9.187633102376982561687969209341, 10.60528123209368064205831515417, 11.36190080701863624502690437072, 12.09576028552389988323422707795

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