Properties

Label 2-224-7.3-c2-0-1
Degree $2$
Conductor $224$
Sign $-0.863 - 0.504i$
Analytic cond. $6.10355$
Root an. cond. $2.47053$
Motivic weight $2$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.863 - 0.504i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ -0.863 - 0.504i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.267597 + 0.988483i\)
\(L(\frac12)\) \(\approx\) \(0.267597 + 0.988483i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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