Properties

Label 2-224-32.5-c1-0-12
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: $\chi_{224} (197, \cdot )$
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

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

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