Properties

Label 2-448-448.171-c1-0-60
Degree $2$
Conductor $448$
Sign $-0.996 - 0.0833i$
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  + (1.12 − 0.860i)2-s + (−2.97 − 1.01i)3-s + (0.520 − 1.93i)4-s + (1.52 − 3.08i)5-s + (−4.21 + 1.42i)6-s + (−0.574 − 2.58i)7-s + (−1.07 − 2.61i)8-s + (5.46 + 4.19i)9-s + (−0.945 − 4.76i)10-s + (1.53 + 1.34i)11-s + (−3.50 + 5.22i)12-s + (−1.18 + 1.78i)13-s + (−2.86 − 2.40i)14-s + (−7.64 + 7.64i)15-s + (−3.45 − 2.00i)16-s + (5.29 − 1.41i)17-s + ⋯
L(s)  = 1  + (0.793 − 0.608i)2-s + (−1.71 − 0.583i)3-s + (0.260 − 0.965i)4-s + (0.680 − 1.37i)5-s + (−1.71 + 0.582i)6-s + (−0.217 − 0.976i)7-s + (−0.380 − 0.924i)8-s + (1.82 + 1.39i)9-s + (−0.299 − 1.50i)10-s + (0.463 + 0.406i)11-s + (−1.01 + 1.50i)12-s + (−0.329 + 0.493i)13-s + (−0.766 − 0.642i)14-s + (−1.97 + 1.97i)15-s + (−0.864 − 0.502i)16-s + (1.28 − 0.343i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.996 - 0.0833i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ -0.996 - 0.0833i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0536296 + 1.28394i\)
\(L(\frac12)\) \(\approx\) \(0.0536296 + 1.28394i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.12 + 0.860i)T \)
7 \( 1 + (0.574 + 2.58i)T \)
good3 \( 1 + (2.97 + 1.01i)T + (2.38 + 1.82i)T^{2} \)
5 \( 1 + (-1.52 + 3.08i)T + (-3.04 - 3.96i)T^{2} \)
11 \( 1 + (-1.53 - 1.34i)T + (1.43 + 10.9i)T^{2} \)
13 \( 1 + (1.18 - 1.78i)T + (-4.97 - 12.0i)T^{2} \)
17 \( 1 + (-5.29 + 1.41i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (6.10 + 0.399i)T + (18.8 + 2.47i)T^{2} \)
23 \( 1 + (-5.26 - 4.04i)T + (5.95 + 22.2i)T^{2} \)
29 \( 1 + (4.44 - 0.884i)T + (26.7 - 11.0i)T^{2} \)
31 \( 1 + (-0.846 - 0.488i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.466 - 0.946i)T + (-22.5 - 29.3i)T^{2} \)
41 \( 1 + (-2.49 + 1.03i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-10.3 - 2.06i)T + (39.7 + 16.4i)T^{2} \)
47 \( 1 + (-0.388 - 0.104i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.16 + 3.60i)T + (-6.91 - 52.5i)T^{2} \)
59 \( 1 + (0.718 + 10.9i)T + (-58.4 + 7.70i)T^{2} \)
61 \( 1 + (-5.44 - 6.20i)T + (-7.96 + 60.4i)T^{2} \)
67 \( 1 + (-3.91 + 11.5i)T + (-53.1 - 40.7i)T^{2} \)
71 \( 1 + (2.42 - 5.85i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (3.89 + 0.512i)T + (70.5 + 18.8i)T^{2} \)
79 \( 1 + (3.28 + 0.880i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (1.87 - 2.80i)T + (-31.7 - 76.6i)T^{2} \)
89 \( 1 + (0.208 + 1.58i)T + (-85.9 + 23.0i)T^{2} \)
97 \( 1 - 5.63T + 97T^{2} \)
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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

−10.91382281863003167088840700591, −10.01963175987296195489929987354, −9.294205430630443954154722565238, −7.40471499725224706577907172318, −6.56333357634779500531521559675, −5.63411287755999916663418892879, −4.94312706221115242329453188370, −4.13900334140141056954729007103, −1.69515357502618373924172481559, −0.813312027357837218511756271637, 2.67512612664094501883117001821, 3.95176741456272857126423649084, 5.28386057490098265355468390725, 5.97754751569408025979691029104, 6.36147059319053014002753427227, 7.34238992250059571064905642121, 8.948925616759005620340462453363, 10.15462673344784208422165980854, 10.80707671439005974771504986819, 11.53441404277698570681943515024

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