Properties

Label 2-224-224.37-c1-0-15
Degree $2$
Conductor $224$
Sign $0.856 + 0.516i$
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  + (−1.13 − 0.846i)2-s + (0.0986 − 0.128i)3-s + (0.565 + 1.91i)4-s + (2.39 − 1.83i)5-s + (−0.220 + 0.0620i)6-s + (0.134 + 2.64i)7-s + (0.983 − 2.65i)8-s + (0.769 + 2.87i)9-s + (−4.26 + 0.0534i)10-s + (−0.230 + 0.0304i)11-s + (0.302 + 0.116i)12-s + (0.451 − 0.186i)13-s + (2.08 − 3.10i)14-s − 0.489i·15-s + (−3.35 + 2.17i)16-s + (5.15 − 2.97i)17-s + ⋯
L(s)  = 1  + (−0.800 − 0.598i)2-s + (0.0569 − 0.0741i)3-s + (0.282 + 0.959i)4-s + (1.07 − 0.821i)5-s + (−0.0900 + 0.0253i)6-s + (0.0507 + 0.998i)7-s + (0.347 − 0.937i)8-s + (0.256 + 0.957i)9-s + (−1.34 + 0.0169i)10-s + (−0.0696 + 0.00916i)11-s + (0.0872 + 0.0336i)12-s + (0.125 − 0.0518i)13-s + (0.557 − 0.830i)14-s − 0.126i·15-s + (−0.839 + 0.542i)16-s + (1.24 − 0.721i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.856 + 0.516i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.856 + 0.516i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00789 - 0.280654i\)
\(L(\frac12)\) \(\approx\) \(1.00789 - 0.280654i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13 + 0.846i)T \)
7 \( 1 + (-0.134 - 2.64i)T \)
good3 \( 1 + (-0.0986 + 0.128i)T + (-0.776 - 2.89i)T^{2} \)
5 \( 1 + (-2.39 + 1.83i)T + (1.29 - 4.82i)T^{2} \)
11 \( 1 + (0.230 - 0.0304i)T + (10.6 - 2.84i)T^{2} \)
13 \( 1 + (-0.451 + 0.186i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + (-5.15 + 2.97i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.456 + 3.46i)T + (-18.3 - 4.91i)T^{2} \)
23 \( 1 + (-0.792 - 2.95i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.01 + 7.27i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (1.65 + 2.86i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.48 + 2.67i)T + (9.57 - 35.7i)T^{2} \)
41 \( 1 + (7.62 - 7.62i)T - 41iT^{2} \)
43 \( 1 + (3.49 - 8.43i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + (3.11 + 1.79i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.617 + 0.0812i)T + (51.1 - 13.7i)T^{2} \)
59 \( 1 + (-1.44 - 10.9i)T + (-56.9 + 15.2i)T^{2} \)
61 \( 1 + (1.36 + 0.179i)T + (58.9 + 15.7i)T^{2} \)
67 \( 1 + (1.13 - 1.47i)T + (-17.3 - 64.7i)T^{2} \)
71 \( 1 + (6.13 + 6.13i)T + 71iT^{2} \)
73 \( 1 + (-5.34 - 1.43i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (12.7 + 7.36i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.17 + 2.14i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-11.0 + 2.95i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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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

−12.02345711909354181113728739733, −11.24410637591950097382678097141, −9.862255518757985617709417132710, −9.481197100200767385805615574470, −8.427810117126984718894578491304, −7.52055730068892720642429925634, −5.88723880653048719509585050936, −4.85064890769338820818774014285, −2.78123400814658041449448351126, −1.60380395966447039800962258710, 1.51307814418116543587059029589, 3.51509706419430527667912349984, 5.43666329284451298505164338744, 6.48937086994238964695688135107, 7.14153408154893086026786922334, 8.385318089357887924396758117097, 9.613492565155751424003429417216, 10.23853311765012942548325741729, 10.80904178410599106917543190131, 12.30407382820529777881867689848

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