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Properties

Label 2-224-224.13-c2-0-18
Degree $2$
Conductor $224$
Sign $0.0846 - 0.996i$
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  + (−1.89 + 0.649i)2-s + (5.08 + 2.10i)3-s + (3.15 − 2.45i)4-s + (0.976 + 2.35i)5-s + (−10.9 − 0.684i)6-s + (−5.09 − 4.79i)7-s + (−4.37 + 6.69i)8-s + (15.0 + 15.0i)9-s + (−3.37 − 3.82i)10-s + (2.51 + 6.07i)11-s + (21.2 − 5.84i)12-s + (−9.18 + 22.1i)13-s + (12.7 + 5.77i)14-s + 14.0i·15-s + (3.93 − 15.5i)16-s + 12.9·17-s + ⋯
L(s)  = 1  + (−0.945 + 0.324i)2-s + (1.69 + 0.702i)3-s + (0.789 − 0.613i)4-s + (0.195 + 0.471i)5-s + (−1.83 − 0.114i)6-s + (−0.727 − 0.685i)7-s + (−0.547 + 0.836i)8-s + (1.67 + 1.67i)9-s + (−0.337 − 0.382i)10-s + (0.228 + 0.552i)11-s + (1.77 − 0.486i)12-s + (−0.706 + 1.70i)13-s + (0.911 + 0.412i)14-s + 0.937i·15-s + (0.246 − 0.969i)16-s + 0.761·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0846 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0846 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.0846 - 0.996i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ 0.0846 - 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.27123 + 1.16785i\)
\(L(\frac12)\) \(\approx\) \(1.27123 + 1.16785i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.89 - 0.649i)T \)
7 \( 1 + (5.09 + 4.79i)T \)
good3 \( 1 + (-5.08 - 2.10i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (-0.976 - 2.35i)T + (-17.6 + 17.6i)T^{2} \)
11 \( 1 + (-2.51 - 6.07i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (9.18 - 22.1i)T + (-119. - 119. i)T^{2} \)
17 \( 1 - 12.9T + 289T^{2} \)
19 \( 1 + (-5.85 + 14.1i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (-3.80 - 3.80i)T + 529iT^{2} \)
29 \( 1 + (-10.4 + 25.3i)T + (-594. - 594. i)T^{2} \)
31 \( 1 - 26.1iT - 961T^{2} \)
37 \( 1 + (22.3 - 9.26i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (1.69 - 1.69i)T - 1.68e3iT^{2} \)
43 \( 1 + (23.2 + 56.0i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 - 63.8T + 2.20e3T^{2} \)
53 \( 1 + (29.7 + 71.8i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (25.6 + 61.8i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (46.9 + 19.4i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (0.333 - 0.805i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-48.5 + 48.5i)T - 5.04e3iT^{2} \)
73 \( 1 + (69.1 - 69.1i)T - 5.32e3iT^{2} \)
79 \( 1 + 104. iT - 6.24e3T^{2} \)
83 \( 1 + (22.3 - 54.0i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-47.9 - 47.9i)T + 7.92e3iT^{2} \)
97 \( 1 + 8.86iT - 9.40e3T^{2} \)
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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

−12.16468055577151041514699815655, −10.70500802944287031720176630484, −9.827171940931683589719776839376, −9.474961515770458508104779034932, −8.524401090796307772138412782480, −7.28266053868799758502390282911, −6.80632539562724215387901689905, −4.61745068854892583306861589581, −3.22498713575502360604555289085, −2.03966523410611645907121630781, 1.15774840543628603739541143781, 2.74021272402213985135633608279, 3.33798913862523419265917447618, 5.88602596004002786209057825796, 7.29877346926864792713017578151, 8.038688130344800871980633423840, 8.849372190640291319637946744200, 9.518704901028103791139883991547, 10.39250833375831530251429477676, 12.23792524515168435944184232120

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