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Properties

Label 2-224-32.3-c2-0-31
Degree $2$
Conductor $224$
Sign $-0.588 + 0.808i$
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  + (−1.03 − 1.71i)2-s + (0.406 − 0.982i)3-s + (−1.87 + 3.53i)4-s + (0.691 + 1.66i)5-s + (−2.10 + 0.315i)6-s + (−1.87 − 1.87i)7-s + (7.98 − 0.433i)8-s + (5.56 + 5.56i)9-s + (2.14 − 2.90i)10-s + (−6.83 − 16.4i)11-s + (2.70 + 3.27i)12-s + (7.71 − 18.6i)13-s + (−1.27 + 5.13i)14-s + 1.92·15-s + (−8.97 − 13.2i)16-s − 1.25i·17-s + ⋯
L(s)  = 1  + (−0.515 − 0.856i)2-s + (0.135 − 0.327i)3-s + (−0.468 + 0.883i)4-s + (0.138 + 0.333i)5-s + (−0.350 + 0.0525i)6-s + (−0.267 − 0.267i)7-s + (0.998 − 0.0541i)8-s + (0.618 + 0.618i)9-s + (0.214 − 0.290i)10-s + (−0.621 − 1.49i)11-s + (0.225 + 0.273i)12-s + (0.593 − 1.43i)13-s + (−0.0912 + 0.366i)14-s + 0.128·15-s + (−0.561 − 0.827i)16-s − 0.0737i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.588 + 0.808i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ -0.588 + 0.808i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.489703 - 0.962671i\)
\(L(\frac12)\) \(\approx\) \(0.489703 - 0.962671i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.03 + 1.71i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good3 \( 1 + (-0.406 + 0.982i)T + (-6.36 - 6.36i)T^{2} \)
5 \( 1 + (-0.691 - 1.66i)T + (-17.6 + 17.6i)T^{2} \)
11 \( 1 + (6.83 + 16.4i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (-7.71 + 18.6i)T + (-119. - 119. i)T^{2} \)
17 \( 1 + 1.25iT - 289T^{2} \)
19 \( 1 + (8.00 + 3.31i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (-4.46 + 4.46i)T - 529iT^{2} \)
29 \( 1 + (15.0 + 6.21i)T + (594. + 594. i)T^{2} \)
31 \( 1 + 1.05iT - 961T^{2} \)
37 \( 1 + (-17.3 - 41.7i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (41.0 + 41.0i)T + 1.68e3iT^{2} \)
43 \( 1 + (18.8 + 45.3i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 16.4T + 2.20e3T^{2} \)
53 \( 1 + (-2.31 + 0.957i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-60.1 + 24.9i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (-111. - 46.3i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (-12.6 + 30.5i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-19.6 - 19.6i)T + 5.04e3iT^{2} \)
73 \( 1 + (-60.5 - 60.5i)T + 5.32e3iT^{2} \)
79 \( 1 + 46.4T + 6.24e3T^{2} \)
83 \( 1 + (12.5 + 5.21i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (31.5 - 31.5i)T - 7.92e3iT^{2} \)
97 \( 1 + 36.2T + 9.40e3T^{2} \)
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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

−11.44412931896687883993126675651, −10.55514148766404615361860555357, −10.17259360628604194872443237886, −8.558952056627369700456327168957, −8.058645009948864775601944817038, −6.81925781261487767156917237525, −5.29825900469378695737445618581, −3.61877945320571728235022685529, −2.56173350545315549108517299495, −0.73090591788205806345096474052, 1.69693721631185057212786719608, 4.11068954034454996498900532902, 5.08022145022648682823818625074, 6.50203116297918621539619101324, 7.21776230444938831161314851829, 8.552243967664778833148099351150, 9.461357386615264949518539763943, 9.915854456695570677938594166963, 11.20192204418527485788002186568, 12.59996308959966858433294495713

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