Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.143 + 0.989i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.89 + 0.649i)2-s + (−5.08 − 2.10i)3-s + (3.15 − 2.45i)4-s + (−0.976 − 2.35i)5-s + (10.9 + 0.684i)6-s + (4.79 + 5.09i)7-s + (−4.37 + 6.69i)8-s + (15.0 + 15.0i)9-s + (3.37 + 3.82i)10-s + (2.51 + 6.07i)11-s + (−21.2 + 5.84i)12-s + (9.18 − 22.1i)13-s + (−12.3 − 6.52i)14-s + 14.0i·15-s + (3.93 − 15.5i)16-s − 12.9·17-s + ⋯
L(s)  = 1  + (−0.945 + 0.324i)2-s + (−1.69 − 0.702i)3-s + (0.789 − 0.613i)4-s + (−0.195 − 0.471i)5-s + (1.83 + 0.114i)6-s + (0.685 + 0.727i)7-s + (−0.547 + 0.836i)8-s + (1.67 + 1.67i)9-s + (0.337 + 0.382i)10-s + (0.228 + 0.552i)11-s + (−1.77 + 0.486i)12-s + (0.706 − 1.70i)13-s + (−0.884 − 0.466i)14-s + 0.937i·15-s + (0.246 − 0.969i)16-s − 0.761·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.143 + 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.143 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.143 + 0.989i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.143 + 0.989i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.294858 - 0.340867i\)
\(L(\frac12)\)  \(\approx\)  \(0.294858 - 0.340867i\)
\(L(2)\)   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 \( 1 + (1.89 - 0.649i)T \)
7 \( 1 + (-4.79 - 5.09i)T \)
good3 \( 1 + (5.08 + 2.10i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (0.976 + 2.35i)T + (-17.6 + 17.6i)T^{2} \)
11 \( 1 + (-2.51 - 6.07i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (-9.18 + 22.1i)T + (-119. - 119. i)T^{2} \)
17 \( 1 + 12.9T + 289T^{2} \)
19 \( 1 + (5.85 - 14.1i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (-3.80 - 3.80i)T + 529iT^{2} \)
29 \( 1 + (-10.4 + 25.3i)T + (-594. - 594. i)T^{2} \)
31 \( 1 + 26.1iT - 961T^{2} \)
37 \( 1 + (22.3 - 9.26i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (-1.69 + 1.69i)T - 1.68e3iT^{2} \)
43 \( 1 + (23.2 + 56.0i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 63.8T + 2.20e3T^{2} \)
53 \( 1 + (29.7 + 71.8i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-25.6 - 61.8i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-46.9 - 19.4i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (0.333 - 0.805i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-48.5 + 48.5i)T - 5.04e3iT^{2} \)
73 \( 1 + (-69.1 + 69.1i)T - 5.32e3iT^{2} \)
79 \( 1 + 104. iT - 6.24e3T^{2} \)
83 \( 1 + (-22.3 + 54.0i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (47.9 + 47.9i)T + 7.92e3iT^{2} \)
97 \( 1 - 8.86iT - 9.40e3T^{2} \)
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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

−11.68791696060851589628806926020, −10.85446833957303514379388275697, −10.07380010784205046780470623425, −8.497555003747778248061194073293, −7.81096577347262049727772418734, −6.57763054340925303036140276382, −5.74537996997723390075077375068, −4.91934488343985292534568373638, −1.84050052761897820818139840968, −0.48201192363159038352676568138, 1.24614333127360899262599829740, 3.76560721921955413177300070455, 4.84013222675545937762630627784, 6.68192489219487578519232747652, 6.76461772467182441990615638447, 8.601249107173771571108623308047, 9.600201663611475648127987261605, 10.81131630112264102440615864467, 11.11654352355519943712546796162, 11.55694816029305676078213438354

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