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Properties

Label 2-224-28.23-c2-0-12
Degree $2$
Conductor $224$
Sign $0.600 + 0.799i$
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  + (4.23 − 2.44i)3-s + (1.16 − 2.00i)5-s + (−4.65 + 5.22i)7-s + (7.42 − 12.8i)9-s + (13.2 − 7.66i)11-s + 14.8·13-s − 11.3i·15-s + (1.19 + 2.07i)17-s + (−25.9 − 14.9i)19-s + (−6.94 + 33.4i)21-s + (−23.0 − 13.2i)23-s + (9.80 + 16.9i)25-s − 28.6i·27-s − 4.75·29-s + (−17.0 + 9.87i)31-s + ⋯
L(s)  = 1  + (1.41 − 0.814i)3-s + (0.232 − 0.401i)5-s + (−0.665 + 0.746i)7-s + (0.825 − 1.42i)9-s + (1.20 − 0.696i)11-s + 1.14·13-s − 0.755i·15-s + (0.0705 + 0.122i)17-s + (−1.36 − 0.787i)19-s + (−0.330 + 1.59i)21-s + (−1.00 − 0.577i)23-s + (0.392 + 0.679i)25-s − 1.05i·27-s − 0.164·29-s + (−0.551 + 0.318i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.600 + 0.799i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ 0.600 + 0.799i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.20946 - 1.10388i\)
\(L(\frac12)\) \(\approx\) \(2.20946 - 1.10388i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (4.65 - 5.22i)T \)
good3 \( 1 + (-4.23 + 2.44i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-1.16 + 2.00i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (-13.2 + 7.66i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 14.8T + 169T^{2} \)
17 \( 1 + (-1.19 - 2.07i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (25.9 + 14.9i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (23.0 + 13.2i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 4.75T + 841T^{2} \)
31 \( 1 + (17.0 - 9.87i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (24.3 - 42.1i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 34.4T + 1.68e3T^{2} \)
43 \( 1 - 59.4iT - 1.84e3T^{2} \)
47 \( 1 + (-32.8 - 18.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-33.2 - 57.6i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (78.2 - 45.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-44.0 + 76.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-15.0 + 8.71i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 23.0iT - 5.04e3T^{2} \)
73 \( 1 + (-34.2 - 59.2i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (54.0 + 31.1i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 36.3iT - 6.88e3T^{2} \)
89 \( 1 + (44.2 - 76.5i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 14.8T + 9.40e3T^{2} \)
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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

−12.25980662856472924883634363374, −10.98655374699820196257258069781, −9.393099973818473708098945204614, −8.831236165609165592161602667102, −8.266955018722431566919899480006, −6.75140847809319096018178160963, −6.03821970732752500240192416035, −3.96695592247059489957818986861, −2.83049238785455048134068137861, −1.44736091188005246314943679390, 2.06629997327496561949769314949, 3.76364633857320158093375293687, 4.00145482524139111924909228905, 6.14869038521947077365311133457, 7.23946788638652104551151489468, 8.461778049233676685340620751759, 9.268353549049093949003325878246, 10.10576499528714008247665452497, 10.77614212129060451709404944142, 12.32253221950399055918167496046

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