Properties

Label 2-224-224.109-c1-0-3
Degree $2$
Conductor $224$
Sign $0.726 - 0.687i$
Analytic cond. $1.78864$
Root an. cond. $1.33740$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.480 + 1.33i)2-s + (−0.758 − 0.988i)3-s + (−1.53 − 1.27i)4-s + (0.171 + 0.131i)5-s + (1.67 − 0.534i)6-s + (1.01 + 2.44i)7-s + (2.43 − 1.43i)8-s + (0.374 − 1.39i)9-s + (−0.256 + 0.164i)10-s + (2.10 + 0.276i)11-s + (−0.0961 + 2.49i)12-s + (3.53 + 1.46i)13-s + (−3.73 + 0.176i)14-s − 0.268i·15-s + (0.734 + 3.93i)16-s + (6.24 + 3.60i)17-s + ⋯
L(s)  = 1  + (−0.339 + 0.940i)2-s + (−0.438 − 0.570i)3-s + (−0.769 − 0.638i)4-s + (0.0765 + 0.0587i)5-s + (0.685 − 0.218i)6-s + (0.383 + 0.923i)7-s + (0.862 − 0.506i)8-s + (0.124 − 0.465i)9-s + (−0.0812 + 0.0520i)10-s + (0.634 + 0.0834i)11-s + (−0.0277 + 0.719i)12-s + (0.980 + 0.406i)13-s + (−0.998 + 0.0471i)14-s − 0.0694i·15-s + (0.183 + 0.982i)16-s + (1.51 + 0.874i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.726 - 0.687i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.726 - 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.888285 + 0.353500i\)
\(L(\frac12)\) \(\approx\) \(0.888285 + 0.353500i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.480 - 1.33i)T \)
7 \( 1 + (-1.01 - 2.44i)T \)
good3 \( 1 + (0.758 + 0.988i)T + (-0.776 + 2.89i)T^{2} \)
5 \( 1 + (-0.171 - 0.131i)T + (1.29 + 4.82i)T^{2} \)
11 \( 1 + (-2.10 - 0.276i)T + (10.6 + 2.84i)T^{2} \)
13 \( 1 + (-3.53 - 1.46i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + (-6.24 - 3.60i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.462 + 3.51i)T + (-18.3 + 4.91i)T^{2} \)
23 \( 1 + (1.34 - 5.02i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.56 + 6.18i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (2.46 - 4.27i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.06 + 1.58i)T + (9.57 + 35.7i)T^{2} \)
41 \( 1 + (-4.31 - 4.31i)T + 41iT^{2} \)
43 \( 1 + (4.15 + 10.0i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + (-4.71 + 2.71i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.20 - 0.817i)T + (51.1 + 13.7i)T^{2} \)
59 \( 1 + (0.990 - 7.52i)T + (-56.9 - 15.2i)T^{2} \)
61 \( 1 + (0.817 - 0.107i)T + (58.9 - 15.7i)T^{2} \)
67 \( 1 + (2.72 + 3.55i)T + (-17.3 + 64.7i)T^{2} \)
71 \( 1 + (2.53 - 2.53i)T - 71iT^{2} \)
73 \( 1 + (11.7 - 3.13i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.02 - 1.16i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.12 + 0.467i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (16.0 + 4.29i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−12.29103136102996264996291438892, −11.61354243322158934862837210109, −10.24715449379402839919671923060, −9.164049563674214346769164482595, −8.384247921090486739112272283531, −7.23936939231585172860687083101, −6.17293406913801435503714586905, −5.62575548419847900032519483642, −3.96982462148214493592145036715, −1.41404452872849767656102253488, 1.30149533186305550350952013544, 3.42807218868649667568348229854, 4.43066247131565615887718523664, 5.59330929475558512488808857999, 7.41154190951474639098431610716, 8.314540549217974292509372008164, 9.583765192847917152805455501481, 10.38010944707407364852358430237, 10.99688743761603753834883276347, 11.85516562787106030402219606295

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