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Properties

Label 2-224-224.125-c0-0-0
Degree $2$
Conductor $224$
Sign $0.831 - 0.555i$
Analytic cond. $0.111790$
Root an. cond. $0.334350$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.707 + 0.292i)11-s + 1.00i·14-s − 1.00·16-s − 1.00·18-s + (0.292 − 0.707i)22-s + (−1 − i)23-s + (−0.707 + 0.707i)25-s + (−0.707 − 0.707i)28-s + (0.707 + 0.292i)29-s + (0.707 − 0.707i)32-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (−0.707 + 0.292i)11-s + 1.00i·14-s − 1.00·16-s − 1.00·18-s + (0.292 − 0.707i)22-s + (−1 − i)23-s + (−0.707 + 0.707i)25-s + (−0.707 − 0.707i)28-s + (0.707 + 0.292i)29-s + (0.707 − 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.831 - 0.555i$
Analytic conductor: \(0.111790\)
Root analytic conductor: \(0.334350\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :0),\ 0.831 - 0.555i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5526385469\)
\(L(\frac12)\) \(\approx\) \(0.5526385469\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-0.707 - 0.707i)T^{2} \)
5 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
13 \( 1 + (0.707 + 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (0.707 + 0.707i)T^{2} \)
23 \( 1 + (1 + i)T + iT^{2} \)
29 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.707 - 0.707i)T^{2} \)
67 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + (0.707 + 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 - T^{2} \)
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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

−12.70719685740070954544584736102, −11.22085608121654554169311210613, −10.46067214844018528815526359447, −9.772348064972584875416691748883, −8.331101550051849999487007881591, −7.69871991938462507386883068052, −6.80257890134922510809578542478, −5.34425991198805266625831792784, −4.38902995657153048539593973232, −1.87327003141636038599724248583, 1.86659789505119545782081617086, 3.42045103322412422864168502433, 4.84629761488409957491405872450, 6.42454372208400273938871110723, 7.81307626176211525269068538871, 8.469677166565118245229208814431, 9.653087339370503317253723176989, 10.32268712461356583066495434374, 11.59556595055060028731739961768, 12.04135720120587371439196385071

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