Properties

Label 2-224-32.11-c2-0-1
Degree $2$
Conductor $224$
Sign $0.686 - 0.727i$
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  + (−1.75 − 0.967i)2-s + (−1.86 − 4.51i)3-s + (2.12 + 3.38i)4-s + (0.599 − 1.44i)5-s + (−1.09 + 9.70i)6-s + (−1.87 + 1.87i)7-s + (−0.453 − 7.98i)8-s + (−10.4 + 10.4i)9-s + (−2.44 + 1.95i)10-s + (−8.17 + 19.7i)11-s + (11.2 − 15.9i)12-s + (1.00 + 2.42i)13-s + (5.08 − 1.46i)14-s − 7.64·15-s + (−6.92 + 14.4i)16-s − 22.9i·17-s + ⋯
L(s)  = 1  + (−0.875 − 0.483i)2-s + (−0.622 − 1.50i)3-s + (0.532 + 0.846i)4-s + (0.119 − 0.289i)5-s + (−0.181 + 1.61i)6-s + (−0.267 + 0.267i)7-s + (−0.0567 − 0.998i)8-s + (−1.16 + 1.16i)9-s + (−0.244 + 0.195i)10-s + (−0.743 + 1.79i)11-s + (0.941 − 1.32i)12-s + (0.0772 + 0.186i)13-s + (0.363 − 0.104i)14-s − 0.509·15-s + (−0.433 + 0.901i)16-s − 1.35i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.686 - 0.727i$
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: \((2,\ 224,\ (\ :1),\ 0.686 - 0.727i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.264542 + 0.114123i\)
\(L(\frac12)\) \(\approx\) \(0.264542 + 0.114123i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.75 + 0.967i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good3 \( 1 + (1.86 + 4.51i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (-0.599 + 1.44i)T + (-17.6 - 17.6i)T^{2} \)
11 \( 1 + (8.17 - 19.7i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (-1.00 - 2.42i)T + (-119. + 119. i)T^{2} \)
17 \( 1 + 22.9iT - 289T^{2} \)
19 \( 1 + (28.2 - 11.6i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (-19.0 - 19.0i)T + 529iT^{2} \)
29 \( 1 + (-23.1 + 9.57i)T + (594. - 594. i)T^{2} \)
31 \( 1 + 12.8iT - 961T^{2} \)
37 \( 1 + (17.0 - 41.2i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + (-16.4 + 16.4i)T - 1.68e3iT^{2} \)
43 \( 1 + (9.75 - 23.5i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + 40.5T + 2.20e3T^{2} \)
53 \( 1 + (79.5 + 32.9i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-61.8 - 25.6i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (85.2 - 35.3i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (7.35 + 17.7i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (1.36 - 1.36i)T - 5.04e3iT^{2} \)
73 \( 1 + (30.6 - 30.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 51.5T + 6.24e3T^{2} \)
83 \( 1 + (16.5 - 6.85i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-35.4 - 35.4i)T + 7.92e3iT^{2} \)
97 \( 1 - 92.1T + 9.40e3T^{2} \)
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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

−12.15998861783305183382286406284, −11.40152957912817757917632108524, −10.22876981888893307046920165286, −9.247670809179661942710476917719, −8.011951451231473345313881792197, −7.20559943042124755408602875740, −6.48296992148419598820064786027, −4.90360943183421066913430565828, −2.57974131693889026223773493928, −1.49118804842394660410175311444, 0.22484757668989381866171659941, 3.08129149207738087957579042240, 4.65741535080742999744093743399, 5.84975161198960270615395268096, 6.53493962975093588612114726342, 8.361622575457845531830767018557, 8.882583229387327331973288963049, 10.24125964455394059389658849825, 10.78066514466735193069387148932, 11.02686385381939175006479459068

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