Properties

Label 2-224-224.69-c2-0-14
Degree $2$
Conductor $224$
Sign $0.560 - 0.828i$
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  + (−1.91 + 0.563i)2-s + (−4.49 + 1.86i)3-s + (3.36 − 2.16i)4-s + (0.432 − 1.04i)5-s + (7.57 − 6.10i)6-s + (−6.93 + 0.951i)7-s + (−5.23 + 6.04i)8-s + (10.3 − 10.3i)9-s + (−0.241 + 2.24i)10-s + (4.35 − 10.5i)11-s + (−11.0 + 15.9i)12-s + (−5.75 − 13.8i)13-s + (12.7 − 5.73i)14-s + 5.49i·15-s + (6.63 − 14.5i)16-s + 2.83·17-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)2-s + (−1.49 + 0.620i)3-s + (0.841 − 0.540i)4-s + (0.0864 − 0.208i)5-s + (1.26 − 1.01i)6-s + (−0.990 + 0.135i)7-s + (−0.654 + 0.756i)8-s + (1.15 − 1.15i)9-s + (−0.0241 + 0.224i)10-s + (0.396 − 0.956i)11-s + (−0.924 + 1.33i)12-s + (−0.442 − 1.06i)13-s + (0.912 − 0.409i)14-s + 0.366i·15-s + (0.414 − 0.909i)16-s + 0.166·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.560 - 0.828i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ 0.560 - 0.828i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.398576 + 0.211597i\)
\(L(\frac12)\) \(\approx\) \(0.398576 + 0.211597i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.91 - 0.563i)T \)
7 \( 1 + (6.93 - 0.951i)T \)
good3 \( 1 + (4.49 - 1.86i)T + (6.36 - 6.36i)T^{2} \)
5 \( 1 + (-0.432 + 1.04i)T + (-17.6 - 17.6i)T^{2} \)
11 \( 1 + (-4.35 + 10.5i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (5.75 + 13.8i)T + (-119. + 119. i)T^{2} \)
17 \( 1 - 2.83T + 289T^{2} \)
19 \( 1 + (-11.1 - 26.9i)T + (-255. + 255. i)T^{2} \)
23 \( 1 + (18.7 - 18.7i)T - 529iT^{2} \)
29 \( 1 + (-0.210 - 0.507i)T + (-594. + 594. i)T^{2} \)
31 \( 1 - 35.8iT - 961T^{2} \)
37 \( 1 + (-59.9 - 24.8i)T + (968. + 968. i)T^{2} \)
41 \( 1 + (20.9 + 20.9i)T + 1.68e3iT^{2} \)
43 \( 1 + (-23.1 + 55.8i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + 2.87T + 2.20e3T^{2} \)
53 \( 1 + (12.1 - 29.2i)T + (-1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-23.7 + 57.4i)T + (-2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (-48.9 + 20.2i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-18.8 - 45.5i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (-40.7 - 40.7i)T + 5.04e3iT^{2} \)
73 \( 1 + (-59.6 - 59.6i)T + 5.32e3iT^{2} \)
79 \( 1 - 89.6iT - 6.24e3T^{2} \)
83 \( 1 + (23.1 + 55.9i)T + (-4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (32.4 - 32.4i)T - 7.92e3iT^{2} \)
97 \( 1 + 29.3iT - 9.40e3T^{2} \)
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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

−11.91147006176413316740260481275, −11.01476465254744205089009779765, −10.10740534932116809866742424693, −9.636211461571967889934790348031, −8.310175627383706784749319150917, −6.96222964532648972006668329991, −5.84232639081122407782899783450, −5.47422360897613714384299571103, −3.42511719029052523760371635828, −0.821299991847207795286261765312, 0.61872696449898657054845784796, 2.36552077815957497370923457144, 4.46482516936937399568601772946, 6.24843162450782304339344900595, 6.72885194246743635574127833493, 7.55947977017298047445033081488, 9.311138239700623435192069543299, 9.942532952714143536900441448209, 11.01908204268664288928899572591, 11.76959080703312963343331512312

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