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Properties

Label 2-448-112.69-c2-0-22
Degree $2$
Conductor $448$
Sign $-0.988 - 0.148i$
Analytic cond. $12.2071$
Root an. cond. $3.49386$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.22 − 2.22i)3-s + (−1.66 + 1.66i)5-s + (4.68 + 5.19i)7-s + 0.878i·9-s + (1.36 + 1.36i)11-s + (−5.09 − 5.09i)13-s + 7.40·15-s − 23.0i·17-s + (−19.4 − 19.4i)19-s + (1.13 − 21.9i)21-s + 7.80i·23-s + 19.4i·25-s + (−18.0 + 18.0i)27-s + (−26.2 + 26.2i)29-s + 21.8i·31-s + ⋯
L(s)  = 1  + (−0.740 − 0.740i)3-s + (−0.333 + 0.333i)5-s + (0.669 + 0.742i)7-s + 0.0976i·9-s + (0.123 + 0.123i)11-s + (−0.391 − 0.391i)13-s + 0.493·15-s − 1.35i·17-s + (−1.02 − 1.02i)19-s + (0.0538 − 1.04i)21-s + 0.339i·23-s + 0.777i·25-s + (−0.668 + 0.668i)27-s + (−0.903 + 0.903i)29-s + 0.705i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.148i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.988 - 0.148i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ -0.988 - 0.148i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1913706758\)
\(L(\frac12)\) \(\approx\) \(0.1913706758\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-4.68 - 5.19i)T \)
good3 \( 1 + (2.22 + 2.22i)T + 9iT^{2} \)
5 \( 1 + (1.66 - 1.66i)T - 25iT^{2} \)
11 \( 1 + (-1.36 - 1.36i)T + 121iT^{2} \)
13 \( 1 + (5.09 + 5.09i)T + 169iT^{2} \)
17 \( 1 + 23.0iT - 289T^{2} \)
19 \( 1 + (19.4 + 19.4i)T + 361iT^{2} \)
23 \( 1 - 7.80iT - 529T^{2} \)
29 \( 1 + (26.2 - 26.2i)T - 841iT^{2} \)
31 \( 1 - 21.8iT - 961T^{2} \)
37 \( 1 + (25.0 + 25.0i)T + 1.36e3iT^{2} \)
41 \( 1 + 64.9T + 1.68e3T^{2} \)
43 \( 1 + (9.70 + 9.70i)T + 1.84e3iT^{2} \)
47 \( 1 + 11.6iT - 2.20e3T^{2} \)
53 \( 1 + (15.1 + 15.1i)T + 2.80e3iT^{2} \)
59 \( 1 + (-14.7 + 14.7i)T - 3.48e3iT^{2} \)
61 \( 1 + (-0.766 - 0.766i)T + 3.72e3iT^{2} \)
67 \( 1 + (50.1 - 50.1i)T - 4.48e3iT^{2} \)
71 \( 1 + 120. iT - 5.04e3T^{2} \)
73 \( 1 - 7.32T + 5.32e3T^{2} \)
79 \( 1 + 7.61T + 6.24e3T^{2} \)
83 \( 1 + (95.8 + 95.8i)T + 6.88e3iT^{2} \)
89 \( 1 + 71.9T + 7.92e3T^{2} \)
97 \( 1 - 121. iT - 9.40e3T^{2} \)
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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

−10.78107902510377776270243535933, −9.403407587495593797308291475236, −8.592837801966682630235907124418, −7.30154337223124537546083125353, −6.89527544597374405528483127997, −5.59693640354828294702706924849, −4.88446284783026042012051912340, −3.19507164442677555095201281845, −1.77614570686099086594292616944, −0.085323696100031399731538617986, 1.80148723283002770523234561004, 3.99429964839032998033774945974, 4.40832155548338568201761757604, 5.55927650636749268027617885978, 6.55867777538537064256006341339, 7.913260998256338506434917059065, 8.468320884536081444568427859705, 9.945530785788370368443211573965, 10.44195112906470145754671835634, 11.26025519461665947283100816027

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