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Properties

Label 2-224-32.13-c1-0-18
Degree $2$
Conductor $224$
Sign $0.999 - 0.00823i$
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  + (0.877 + 1.10i)2-s + (0.965 − 2.33i)3-s + (−0.460 + 1.94i)4-s + (1.27 − 0.527i)5-s + (3.43 − 0.974i)6-s + (0.707 − 0.707i)7-s + (−2.56 + 1.19i)8-s + (−2.37 − 2.37i)9-s + (1.70 + 0.950i)10-s + (−0.151 − 0.365i)11-s + (4.09 + 2.95i)12-s + (−3.55 − 1.47i)13-s + (1.40 + 0.163i)14-s − 3.48i·15-s + (−3.57 − 1.79i)16-s + 6.40i·17-s + ⋯
L(s)  = 1  + (0.620 + 0.784i)2-s + (0.557 − 1.34i)3-s + (−0.230 + 0.973i)4-s + (0.570 − 0.236i)5-s + (1.40 − 0.397i)6-s + (0.267 − 0.267i)7-s + (−0.906 + 0.423i)8-s + (−0.793 − 0.793i)9-s + (0.538 + 0.300i)10-s + (−0.0456 − 0.110i)11-s + (1.18 + 0.852i)12-s + (−0.985 − 0.408i)13-s + (0.375 + 0.0438i)14-s − 0.898i·15-s + (−0.893 − 0.448i)16-s + 1.55i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.999 - 0.00823i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.999 - 0.00823i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97181 + 0.00812301i\)
\(L(\frac12)\) \(\approx\) \(1.97181 + 0.00812301i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.877 - 1.10i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-0.965 + 2.33i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (-1.27 + 0.527i)T + (3.53 - 3.53i)T^{2} \)
11 \( 1 + (0.151 + 0.365i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (3.55 + 1.47i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 6.40iT - 17T^{2} \)
19 \( 1 + (-7.26 - 3.00i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (2.60 + 2.60i)T + 23iT^{2} \)
29 \( 1 + (0.194 - 0.469i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 3.21T + 31T^{2} \)
37 \( 1 + (6.38 - 2.64i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (6.46 + 6.46i)T + 41iT^{2} \)
43 \( 1 + (1.24 + 3.00i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 6.28iT - 47T^{2} \)
53 \( 1 + (-3.30 - 7.97i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-2.65 + 1.10i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-3.76 + 9.09i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (0.104 - 0.251i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (0.0676 - 0.0676i)T - 71iT^{2} \)
73 \( 1 + (3.14 + 3.14i)T + 73iT^{2} \)
79 \( 1 + 13.6iT - 79T^{2} \)
83 \( 1 + (-8.23 - 3.41i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (2.87 - 2.87i)T - 89iT^{2} \)
97 \( 1 - 17.3T + 97T^{2} \)
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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

−12.52517971466452177971043615681, −11.93731660082686585485771844323, −10.20973440045285731547544226352, −8.863440286881473686418964267468, −7.893789380195333443347179396132, −7.33700078913311745538545862782, −6.18252095467720137008085494098, −5.21215495826024667963490041343, −3.47513255408202093127261298271, −1.90236329765249158364001681299, 2.37878088874498972758759075217, 3.43335028268765979413311145985, 4.80251185092719484523885645933, 5.37998704634562095173094229041, 7.12368333814685058485603054710, 8.907406492168789710762874424761, 9.812554029010326295722710867485, 9.952910079741185010399565684711, 11.39464330600793657266878635409, 11.93945741062908145008292689349

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