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Properties

Label 2-224-56.19-c1-0-1
Degree $2$
Conductor $224$
Sign $0.261 - 0.965i$
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  + (0.416 − 0.240i)3-s + (−1.59 + 2.76i)5-s + (0.694 + 2.55i)7-s + (−1.38 + 2.39i)9-s + (−0.800 − 1.38i)11-s + 1.38·13-s + 1.53i·15-s + (3.48 − 2.01i)17-s + (4.56 + 2.63i)19-s + (0.902 + 0.896i)21-s + (−3.83 − 2.21i)23-s + (−2.60 − 4.50i)25-s + 2.77i·27-s − 5.10i·29-s + (−0.0579 − 0.100i)31-s + ⋯
L(s)  = 1  + (0.240 − 0.138i)3-s + (−0.714 + 1.23i)5-s + (0.262 + 0.964i)7-s + (−0.461 + 0.799i)9-s + (−0.241 − 0.418i)11-s + 0.385·13-s + 0.396i·15-s + (0.845 − 0.488i)17-s + (1.04 + 0.605i)19-s + (0.197 + 0.195i)21-s + (−0.798 − 0.461i)23-s + (−0.520 − 0.901i)25-s + 0.533i·27-s − 0.948i·29-s + (−0.0104 − 0.0180i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.261 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.261 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.261 - 0.965i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.261 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.898333 + 0.687446i\)
\(L(\frac12)\) \(\approx\) \(0.898333 + 0.687446i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.694 - 2.55i)T \)
good3 \( 1 + (-0.416 + 0.240i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.59 - 2.76i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.800 + 1.38i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.38T + 13T^{2} \)
17 \( 1 + (-3.48 + 2.01i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.56 - 2.63i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.83 + 2.21i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.10iT - 29T^{2} \)
31 \( 1 + (0.0579 + 0.100i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.63 - 2.67i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.21iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-5.05 + 8.76i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.13 - 3.54i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.38 + 2.53i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.21 + 7.29i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.01 + 8.69i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.29iT - 71T^{2} \)
73 \( 1 + (-9.30 + 5.37i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-10.3 - 5.96i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.9iT - 83T^{2} \)
89 \( 1 + (-1.5 - 0.866i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.87iT - 97T^{2} \)
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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

−12.16413808581205562971454073710, −11.50079469309895687383886205904, −10.72038093017733811430131150600, −9.583592576139686180893527440892, −8.110636236074100152551126509673, −7.79173045611743675783368522357, −6.32842813300626629175327436420, −5.26728146361849721858839195256, −3.47327548874968975347129256901, −2.47470744104478873413309340481, 1.00421831295850603881446398270, 3.50227978693450598204136909361, 4.44292851726025047692301044283, 5.65615226713392772345165564950, 7.27816872696351385245326325900, 8.111755970513680294500744238083, 9.042173927610224213519392698340, 9.988495037979147927772812712816, 11.24430996156829338786853522426, 12.10515518435380573705680968811

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