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Properties

Label 2-224-32.11-c2-0-14
Degree $2$
Conductor $224$
Sign $0.831 - 0.555i$
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  + (−1.70 − 1.04i)2-s + (0.502 + 1.21i)3-s + (1.81 + 3.56i)4-s + (−0.0438 + 0.105i)5-s + (0.410 − 2.59i)6-s + (−1.87 + 1.87i)7-s + (0.626 − 7.97i)8-s + (5.14 − 5.14i)9-s + (0.185 − 0.134i)10-s + (1.37 − 3.31i)11-s + (−3.41 + 3.99i)12-s + (5.53 + 13.3i)13-s + (5.14 − 1.23i)14-s − 0.150·15-s + (−9.40 + 12.9i)16-s + 25.0i·17-s + ⋯
L(s)  = 1  + (−0.852 − 0.522i)2-s + (0.167 + 0.404i)3-s + (0.454 + 0.890i)4-s + (−0.00877 + 0.0211i)5-s + (0.0684 − 0.432i)6-s + (−0.267 + 0.267i)7-s + (0.0782 − 0.996i)8-s + (0.571 − 0.571i)9-s + (0.0185 − 0.0134i)10-s + (0.124 − 0.301i)11-s + (−0.284 + 0.333i)12-s + (0.426 + 1.02i)13-s + (0.367 − 0.0882i)14-s − 0.0100·15-s + (−0.587 + 0.809i)16-s + 1.47i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.831 - 0.555i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ 0.831 - 0.555i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.05097 + 0.319006i\)
\(L(\frac12)\) \(\approx\) \(1.05097 + 0.319006i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.70 + 1.04i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good3 \( 1 + (-0.502 - 1.21i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (0.0438 - 0.105i)T + (-17.6 - 17.6i)T^{2} \)
11 \( 1 + (-1.37 + 3.31i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (-5.53 - 13.3i)T + (-119. + 119. i)T^{2} \)
17 \( 1 - 25.0iT - 289T^{2} \)
19 \( 1 + (-4.85 + 2.01i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (1.30 + 1.30i)T + 529iT^{2} \)
29 \( 1 + (-15.1 + 6.25i)T + (594. - 594. i)T^{2} \)
31 \( 1 - 41.0iT - 961T^{2} \)
37 \( 1 + (9.38 - 22.6i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + (-33.4 + 33.4i)T - 1.68e3iT^{2} \)
43 \( 1 + (-4.08 + 9.87i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 20.9T + 2.20e3T^{2} \)
53 \( 1 + (-71.4 - 29.6i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (84.5 + 35.0i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (35.0 - 14.5i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (34.1 + 82.4i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (26.1 - 26.1i)T - 5.04e3iT^{2} \)
73 \( 1 + (22.9 - 22.9i)T - 5.32e3iT^{2} \)
79 \( 1 + 120.T + 6.24e3T^{2} \)
83 \( 1 + (2.08 - 0.863i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-49.1 - 49.1i)T + 7.92e3iT^{2} \)
97 \( 1 - 123.T + 9.40e3T^{2} \)
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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

−12.05771965743394723895375511009, −10.91206183460070619568150274855, −10.18684154218558877378872348323, −9.096417804637876297849718616159, −8.642183977285892999222927016828, −7.17783390470410976307797477409, −6.23692311816649720448337672062, −4.24823082508846316723315866757, −3.23152712903352715243752321284, −1.45929166131302571762511113183, 0.888346005625389161049702079289, 2.61733253595290375890623247252, 4.72484354006270204831511290781, 6.01601087610402432823367389795, 7.21146597042369975716763164948, 7.75794211849941845198774976488, 8.909461241852193662339883281729, 9.952670571595173148668916503233, 10.66062589228161254823765439524, 11.81066782735790587063007885670

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