Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.863 + 0.504i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.45 − 1.99i)3-s + (−7.80 − 4.50i)5-s + (−5.54 + 4.26i)7-s + (3.44 − 5.96i)9-s + (−8.28 − 14.3i)11-s − 0.446i·13-s − 35.9·15-s + (6.02 − 3.47i)17-s + (−11.0 − 6.37i)19-s + (−10.6 + 25.7i)21-s + (13.2 − 23.0i)23-s + (28.1 + 48.7i)25-s + 8.43i·27-s + 26.4·29-s + (−21.7 + 12.5i)31-s + ⋯
L(s)  = 1  + (1.15 − 0.664i)3-s + (−1.56 − 0.901i)5-s + (−0.792 + 0.609i)7-s + (0.382 − 0.662i)9-s + (−0.752 − 1.30i)11-s − 0.0343i·13-s − 2.39·15-s + (0.354 − 0.204i)17-s + (−0.580 − 0.335i)19-s + (−0.507 + 1.22i)21-s + (0.577 − 1.00i)23-s + (1.12 + 1.95i)25-s + 0.312i·27-s + 0.912·29-s + (−0.702 + 0.405i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 + 0.504i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.863 + 0.504i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (33, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.863 + 0.504i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.267597 - 0.988483i\)
\(L(\frac12)\)  \(\approx\)  \(0.267597 - 0.988483i\)
\(L(2)\)   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 \( 1 + (5.54 - 4.26i)T \)
good3 \( 1 + (-3.45 + 1.99i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (7.80 + 4.50i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (8.28 + 14.3i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 0.446iT - 169T^{2} \)
17 \( 1 + (-6.02 + 3.47i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (11.0 + 6.37i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-13.2 + 23.0i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 26.4T + 841T^{2} \)
31 \( 1 + (21.7 - 12.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-31.6 + 54.9i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 0.519iT - 1.68e3T^{2} \)
43 \( 1 + 25.5T + 1.84e3T^{2} \)
47 \( 1 + (59.4 + 34.3i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-3.58 - 6.20i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-65.3 + 37.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-39.8 - 23.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (21.4 + 37.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 60.0T + 5.04e3T^{2} \)
73 \( 1 + (40.5 - 23.3i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-27.1 + 47.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 11.4iT - 6.88e3T^{2} \)
89 \( 1 + (-53.1 - 30.7i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 20.3iT - 9.40e3T^{2} \)
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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

−11.87213616373888233460919798141, −10.81347820341945809827245562066, −9.137651246546468902640246604867, −8.473160581688970779431920359899, −7.989649771506669887837411718688, −6.81557005301462538839215336494, −5.21563874579977730601824837063, −3.66830784890102636952925532821, −2.73718027712871353251950662767, −0.47547215199922233592959120398, 2.82235296501782338640749084961, 3.64689712568212471329915685833, 4.51910106268749620952984272529, 6.72839458304564165331266229039, 7.60077458896500217858010271673, 8.289434786076996195836049739084, 9.724509748412069159112690895960, 10.25494604711460896298116458707, 11.34131954789475506359910484069, 12.46085652046216106248196717186

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