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Properties

Label 2-448-448.227-c1-0-18
Degree $2$
Conductor $448$
Sign $0.0635 - 0.997i$
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  + (−1.03 + 0.959i)2-s + (0.896 − 1.02i)3-s + (0.157 − 1.99i)4-s + (−0.0641 + 0.978i)5-s + (0.0501 + 1.92i)6-s + (−0.310 + 2.62i)7-s + (1.75 + 2.22i)8-s + (0.150 + 1.14i)9-s + (−0.872 − 1.07i)10-s + (−0.0180 + 0.0532i)11-s + (−1.89 − 1.94i)12-s + (−0.0490 + 0.0327i)13-s + (−2.19 − 3.02i)14-s + (0.942 + 0.942i)15-s + (−3.95 − 0.627i)16-s + (0.457 + 1.70i)17-s + ⋯
L(s)  = 1  + (−0.734 + 0.678i)2-s + (0.517 − 0.590i)3-s + (0.0786 − 0.996i)4-s + (−0.0286 + 0.437i)5-s + (0.0204 + 0.784i)6-s + (−0.117 + 0.993i)7-s + (0.618 + 0.785i)8-s + (0.0500 + 0.380i)9-s + (−0.275 − 0.340i)10-s + (−0.00545 + 0.0160i)11-s + (−0.547 − 0.562i)12-s + (−0.0135 + 0.00908i)13-s + (−0.587 − 0.809i)14-s + (0.243 + 0.243i)15-s + (−0.987 − 0.156i)16-s + (0.110 + 0.413i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0635 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0635 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.0635 - 0.997i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.0635 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.770770 + 0.723249i\)
\(L(\frac12)\) \(\approx\) \(0.770770 + 0.723249i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.03 - 0.959i)T \)
7 \( 1 + (0.310 - 2.62i)T \)
good3 \( 1 + (-0.896 + 1.02i)T + (-0.391 - 2.97i)T^{2} \)
5 \( 1 + (0.0641 - 0.978i)T + (-4.95 - 0.652i)T^{2} \)
11 \( 1 + (0.0180 - 0.0532i)T + (-8.72 - 6.69i)T^{2} \)
13 \( 1 + (0.0490 - 0.0327i)T + (4.97 - 12.0i)T^{2} \)
17 \( 1 + (-0.457 - 1.70i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.12 + 1.53i)T + (11.5 + 15.0i)T^{2} \)
23 \( 1 + (-0.541 - 4.11i)T + (-22.2 + 5.95i)T^{2} \)
29 \( 1 + (0.832 - 4.18i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + (-2.45 - 1.41i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.188 - 2.87i)T + (-36.6 - 4.82i)T^{2} \)
41 \( 1 + (-6.51 - 2.69i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (1.62 + 8.17i)T + (-39.7 + 16.4i)T^{2} \)
47 \( 1 + (0.754 - 2.81i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-12.7 - 4.32i)T + (42.0 + 32.2i)T^{2} \)
59 \( 1 + (3.63 + 7.36i)T + (-35.9 + 46.8i)T^{2} \)
61 \( 1 + (6.78 - 2.30i)T + (48.3 - 37.1i)T^{2} \)
67 \( 1 + (3.64 + 3.20i)T + (8.74 + 66.4i)T^{2} \)
71 \( 1 + (1.97 + 4.75i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (6.45 + 8.40i)T + (-18.8 + 70.5i)T^{2} \)
79 \( 1 + (0.188 - 0.704i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (-6.23 + 4.16i)T + (31.7 - 76.6i)T^{2} \)
89 \( 1 + (-4.33 - 3.32i)T + (23.0 + 85.9i)T^{2} \)
97 \( 1 + 7.78T + 97T^{2} \)
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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

−11.04492324593423664928090895801, −10.34651742423744637396866701866, −9.144011851811721234181792671287, −8.593087297349818785493218502939, −7.67764477600103670207842081202, −6.88179394995412131564600260895, −5.94835204012919374329234443913, −4.87833062462667663814684060399, −2.87907783556301170273111599275, −1.74253576884976477922110921123, 0.833655868946705074419145566890, 2.66604904229180349510880836774, 3.87845760563729105542473991162, 4.52856496039297679424357598727, 6.44643630039224303597352264299, 7.49579984444624221574116757292, 8.468004553950524403187278792396, 9.154046854416754074889156067177, 10.04058149979626651822383636032, 10.56440517230005927403172121733

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