Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.935 - 0.353i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.585·3-s + 9.03i·5-s − 2.64i·7-s − 8.65·9-s − 12.4·11-s − 9.03i·13-s − 5.29i·15-s + 12.3·17-s − 28.8·19-s + 1.54i·21-s + 24.6i·23-s − 56.5·25-s + 10.3·27-s + 22.4i·29-s + 16.7i·31-s + ⋯
L(s)  = 1  − 0.195·3-s + 1.80i·5-s − 0.377i·7-s − 0.961·9-s − 1.13·11-s − 0.694i·13-s − 0.352i·15-s + 0.726·17-s − 1.51·19-s + 0.0738i·21-s + 1.07i·23-s − 2.26·25-s + 0.383·27-s + 0.774i·29-s + 0.541i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.935 - 0.353i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (15, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.935 - 0.353i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.110935 + 0.607279i\)
\(L(\frac12)\)  \(\approx\)  \(0.110935 + 0.607279i\)
\(L(2)\)   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 \( 1 + 0.585T + 9T^{2} \)
5 \( 1 - 9.03iT - 25T^{2} \)
11 \( 1 + 12.4T + 121T^{2} \)
13 \( 1 + 9.03iT - 169T^{2} \)
17 \( 1 - 12.3T + 289T^{2} \)
19 \( 1 + 28.8T + 361T^{2} \)
23 \( 1 - 24.6iT - 529T^{2} \)
29 \( 1 - 22.4iT - 841T^{2} \)
31 \( 1 - 16.7iT - 961T^{2} \)
37 \( 1 - 16.2iT - 1.36e3T^{2} \)
41 \( 1 - 6.97T + 1.68e3T^{2} \)
43 \( 1 - 22.8T + 1.84e3T^{2} \)
47 \( 1 - 6.19iT - 2.20e3T^{2} \)
53 \( 1 - 8.01iT - 2.80e3T^{2} \)
59 \( 1 + 30.4T + 3.48e3T^{2} \)
61 \( 1 + 15.2iT - 3.72e3T^{2} \)
67 \( 1 - 78.6T + 4.48e3T^{2} \)
71 \( 1 - 17.5iT - 5.04e3T^{2} \)
73 \( 1 - 46.6T + 5.32e3T^{2} \)
79 \( 1 - 81.0iT - 6.24e3T^{2} \)
83 \( 1 + 40.3T + 6.88e3T^{2} \)
89 \( 1 - 111.T + 7.92e3T^{2} \)
97 \( 1 + 164.T + 9.40e3T^{2} \)
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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

−12.38613350417916851149909192443, −11.05749074188543220093535889376, −10.76613386397658514879895170958, −9.901230113248688483969989513604, −8.260404861765974515613790328544, −7.41245061528077777686058289450, −6.35926626301545781119199235191, −5.38204713638352848394000394657, −3.47947332509970968791082260630, −2.58025481637514242409427015417, 0.32103442146383376525270727863, 2.27441102717383305559999103030, 4.30917080448361251093015817601, 5.26355230200992111137123162311, 6.12774055583072743141955483198, 7.993663960780479254245842136764, 8.569815113634751333028320854399, 9.433433574077733217975255543543, 10.69509529813213666595694561870, 11.82813430929697822503950407852

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