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Properties

Label 2-448-112.109-c1-0-3
Degree $2$
Conductor $448$
Sign $0.917 + 0.398i$
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  + (−2.51 + 0.672i)3-s + (−2.91 − 0.780i)5-s + (−1.41 + 2.23i)7-s + (3.25 − 1.87i)9-s + (−0.838 − 3.12i)11-s + (2.52 + 2.52i)13-s + 7.83·15-s + (0.201 − 0.348i)17-s + (0.373 − 1.39i)19-s + (2.06 − 6.56i)21-s + (7.89 − 4.55i)23-s + (3.54 + 2.04i)25-s + (−1.39 + 1.39i)27-s + (1.47 + 1.47i)29-s + (−2.12 + 3.68i)31-s + ⋯
L(s)  = 1  + (−1.44 + 0.388i)3-s + (−1.30 − 0.348i)5-s + (−0.536 + 0.843i)7-s + (1.08 − 0.626i)9-s + (−0.252 − 0.943i)11-s + (0.699 + 0.699i)13-s + 2.02·15-s + (0.0487 − 0.0844i)17-s + (0.0855 − 0.319i)19-s + (0.449 − 1.43i)21-s + (1.64 − 0.949i)23-s + (0.708 + 0.408i)25-s + (−0.267 + 0.267i)27-s + (0.273 + 0.273i)29-s + (−0.381 + 0.661i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.917 + 0.398i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.917 + 0.398i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.498226 - 0.103531i\)
\(L(\frac12)\) \(\approx\) \(0.498226 - 0.103531i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 - 2.23i)T \)
good3 \( 1 + (2.51 - 0.672i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (2.91 + 0.780i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.838 + 3.12i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-2.52 - 2.52i)T + 13iT^{2} \)
17 \( 1 + (-0.201 + 0.348i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.373 + 1.39i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-7.89 + 4.55i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.47 - 1.47i)T + 29iT^{2} \)
31 \( 1 + (2.12 - 3.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.94 - 0.520i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 8.96iT - 41T^{2} \)
43 \( 1 + (-0.997 + 0.997i)T - 43iT^{2} \)
47 \( 1 + (-2.09 - 3.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.488 - 1.82i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.636 - 2.37i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.685 - 2.55i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-11.6 + 3.12i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.451iT - 71T^{2} \)
73 \( 1 + (9.40 + 5.43i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.31 + 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.742 + 0.742i)T + 83iT^{2} \)
89 \( 1 + (-11.1 + 6.41i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.3T + 97T^{2} \)
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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

−11.11831717888367077480660851443, −10.57997247472946438216597865548, −9.073396626349371837065552791549, −8.542617516781873079572696560535, −7.11990174098973755842453818164, −6.21688261127682349468531246832, −5.30309457478544190827317750755, −4.39369788447929970664049695471, −3.19645256927197639194636816229, −0.56486427451961607747661403992, 0.897578756446565683485041429528, 3.34209812670998242035932979813, 4.40038335943765365694482965521, 5.50818125673863347396582123429, 6.65871807067704698317320381402, 7.28479349274456995472436298176, 8.023616665361384025234712765990, 9.656449527849225281512049897567, 10.62514496455821480420561016261, 11.20076646514558921185974365127

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