Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.664 - 0.746i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.438 + 0.253i)3-s + (−4.59 − 2.65i)5-s + (5.27 + 4.59i)7-s + (−4.37 + 7.57i)9-s + (−8.54 − 14.8i)11-s + 21.4i·13-s + 2.68·15-s + (−20.7 + 12.0i)17-s + (10.5 + 6.11i)19-s + (−3.48 − 0.680i)21-s + (−20.1 + 34.8i)23-s + (1.55 + 2.69i)25-s − 8.99i·27-s − 26.0·29-s + (21.8 − 12.6i)31-s + ⋯
L(s)  = 1  + (−0.146 + 0.0844i)3-s + (−0.918 − 0.530i)5-s + (0.753 + 0.656i)7-s + (−0.485 + 0.841i)9-s + (−0.776 − 1.34i)11-s + 1.65i·13-s + 0.179·15-s + (−1.22 + 0.706i)17-s + (0.557 + 0.321i)19-s + (−0.165 − 0.0324i)21-s + (−0.875 + 1.51i)23-s + (0.0623 + 0.107i)25-s − 0.333i·27-s − 0.899·29-s + (0.705 − 0.407i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.746i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.664 - 0.746i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (33, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.664 - 0.746i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.245743 + 0.547724i\)
\(L(\frac12)\)  \(\approx\)  \(0.245743 + 0.547724i\)
\(L(2)\)   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 \( 1 + (-5.27 - 4.59i)T \)
good3 \( 1 + (0.438 - 0.253i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (4.59 + 2.65i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (8.54 + 14.8i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 21.4iT - 169T^{2} \)
17 \( 1 + (20.7 - 12.0i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-10.5 - 6.11i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (20.1 - 34.8i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 26.0T + 841T^{2} \)
31 \( 1 + (-21.8 + 12.6i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (6.48 - 11.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 33.8iT - 1.68e3T^{2} \)
43 \( 1 + 29.9T + 1.84e3T^{2} \)
47 \( 1 + (-48.2 - 27.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-4.36 - 7.55i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-43.2 + 24.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-3.40 - 1.96i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (52.9 + 91.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 35.0T + 5.04e3T^{2} \)
73 \( 1 + (-40.3 + 23.3i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-43.4 + 75.3i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 64.0iT - 6.88e3T^{2} \)
89 \( 1 + (-37.2 - 21.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 28.7iT - 9.40e3T^{2} \)
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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

−12.00853441871247788080346344743, −11.47505191882750033304677198194, −10.83673778199010834764821302627, −9.179629689341758504120658372540, −8.344045491027752631950122790851, −7.73607650406413521789718087867, −6.02423421474527707948528004188, −5.03046300159555583780782379236, −3.89708458836309901968278693418, −2.05407864261556472736598683598, 0.31997725516089513036062134426, 2.69501997856623799487442414189, 4.11720271284336151590300439400, 5.23615449698256272429522902298, 6.84165260473692902591601652207, 7.58374165620684751288278194503, 8.463156257939130498992852102930, 9.971205561746563222796573783534, 10.78694465865824073368271209467, 11.62850543462351444250235062034

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