Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.624 - 0.780i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.94 + 3.37i)3-s + (4.42 + 7.67i)5-s + (6.92 + 1.03i)7-s + (−3.09 − 5.35i)9-s + (−3.15 − 1.82i)11-s + 7.79·13-s − 34.5·15-s + (−9.07 − 5.23i)17-s + (5.39 + 9.34i)19-s + (−16.9 + 21.3i)21-s + (−6.45 − 11.1i)23-s + (−26.7 + 46.3i)25-s − 10.9·27-s − 17.2i·29-s + (26.1 + 15.1i)31-s + ⋯
L(s)  = 1  + (−0.649 + 1.12i)3-s + (0.885 + 1.53i)5-s + (0.989 + 0.147i)7-s + (−0.343 − 0.594i)9-s + (−0.287 − 0.165i)11-s + 0.599·13-s − 2.30·15-s + (−0.533 − 0.308i)17-s + (0.283 + 0.491i)19-s + (−0.808 + 1.01i)21-s + (−0.280 − 0.486i)23-s + (−1.06 + 1.85i)25-s − 0.406·27-s − 0.594i·29-s + (0.844 + 0.487i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 - 0.780i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.624 - 0.780i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.624 - 0.780i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.661117 + 1.37551i\)
\(L(\frac12)\)  \(\approx\)  \(0.661117 + 1.37551i\)
\(L(2)\)   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 \( 1 + (-6.92 - 1.03i)T \)
good3 \( 1 + (1.94 - 3.37i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (-4.42 - 7.67i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (3.15 + 1.82i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 7.79T + 169T^{2} \)
17 \( 1 + (9.07 + 5.23i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-5.39 - 9.34i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (6.45 + 11.1i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 17.2iT - 841T^{2} \)
31 \( 1 + (-26.1 - 15.1i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (34.2 - 19.7i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 73.6iT - 1.68e3T^{2} \)
43 \( 1 + 40.8iT - 1.84e3T^{2} \)
47 \( 1 + (-36.2 + 20.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-5.55 - 3.20i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (7.95 - 13.7i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-6.07 - 10.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-6.75 - 3.89i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 41.3T + 5.04e3T^{2} \)
73 \( 1 + (-77.6 - 44.8i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-35.3 - 61.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 60.8T + 6.88e3T^{2} \)
89 \( 1 + (23.4 - 13.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 3.26iT - 9.40e3T^{2} \)
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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

−11.95954374500140801092200697574, −11.02359923871252581805124108751, −10.56253042167163769563286756225, −9.878495882629170372643050906789, −8.586579894547197736682404190938, −7.15472423183315861778585005969, −6.00281470913398900527391530365, −5.17626284095366537817463498566, −3.80009989188494090503579958858, −2.26656177277850376116851572860, 0.983074735940171440461394743583, 1.87214536882345445114836127537, 4.56322378874166447783519202268, 5.47100669579471356729207350917, 6.39664767091496698404825910258, 7.73864893410059294085698452331, 8.577349323784700161664620495260, 9.609652930638876376723764908214, 10.98668745318946965677085103991, 11.86707845884149753105343448118

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