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Properties

Label 2-448-56.3-c1-0-1
Degree $2$
Conductor $448$
Sign $0.319 - 0.947i$
Analytic cond. $3.57729$
Root an. cond. $1.89137$
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  + (−1.93 − 1.11i)3-s + (−1.11 − 1.93i)5-s + (−1.73 + 2i)7-s + (1 + 1.73i)9-s + (−1.93 + 3.35i)11-s + 5.00i·15-s + (4.5 + 2.59i)17-s + (5.80 − 3.35i)19-s + (5.59 − 1.93i)21-s + (−2.59 + 1.5i)23-s + 2.23i·27-s + 7.74i·29-s + (−0.866 + 1.5i)31-s + (7.50 − 4.33i)33-s + (5.80 + 1.11i)35-s + ⋯
L(s)  = 1  + (−1.11 − 0.645i)3-s + (−0.499 − 0.866i)5-s + (−0.654 + 0.755i)7-s + (0.333 + 0.577i)9-s + (−0.583 + 1.01i)11-s + 1.29i·15-s + (1.09 + 0.630i)17-s + (1.33 − 0.769i)19-s + (1.21 − 0.422i)21-s + (−0.541 + 0.312i)23-s + 0.430i·27-s + 1.43i·29-s + (−0.155 + 0.269i)31-s + (1.30 − 0.753i)33-s + (0.981 + 0.188i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.319 - 0.947i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.319 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.362133 + 0.260069i\)
\(L(\frac12)\) \(\approx\) \(0.362133 + 0.260069i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (1.73 - 2i)T \)
good3 \( 1 + (1.93 + 1.11i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.11 + 1.93i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.93 - 3.35i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-4.5 - 2.59i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.80 + 3.35i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.59 - 1.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.74iT - 29T^{2} \)
31 \( 1 + (0.866 - 1.5i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (10.0 - 5.80i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (2.59 + 4.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.35 - 1.93i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.93 + 1.11i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.35 + 5.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.80 - 10.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + (4.5 + 2.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-11.2 + 6.5i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.47iT - 83T^{2} \)
89 \( 1 + (4.5 - 2.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.46iT - 97T^{2} \)
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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

−11.68509960266859622937934669173, −10.42163649423646190802205952742, −9.536231580233326708785350907076, −8.496534626251850545819486303727, −7.46064019421813837238416949058, −6.60433239736894912690548840323, −5.44519254395064145253920696131, −4.96280166596327501392016016270, −3.24316805434977761924415275522, −1.36312896730591228081064368954, 0.34666289553865611288580353702, 3.13041585379470395460645242294, 3.92110615410783594690340378474, 5.36900710000439539521892928725, 6.01244902131811409187268621240, 7.20702021807699754313436229153, 7.908991717074370668343738809875, 9.507396218805077449242193573008, 10.36106413456293362908846175490, 10.75098604549229115964437759883

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