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Properties

Label 2-224-224.109-c1-0-9
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: $\chi_{224} (109, \cdot )$
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

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

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