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Properties

Label 2-448-7.4-c1-0-4
Degree $2$
Conductor $448$
Sign $0.386 - 0.922i$
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  + (−0.5 + 0.866i)3-s + (1.5 + 2.59i)5-s + (2 − 1.73i)7-s + (1 + 1.73i)9-s + (1.5 − 2.59i)11-s − 2·13-s − 3·15-s + (−1.5 + 2.59i)17-s + (0.5 + 0.866i)19-s + (0.499 + 2.59i)21-s + (1.5 + 2.59i)23-s + (−2 + 3.46i)25-s − 5·27-s + 6·29-s + (−3.5 + 6.06i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.670 + 1.16i)5-s + (0.755 − 0.654i)7-s + (0.333 + 0.577i)9-s + (0.452 − 0.783i)11-s − 0.554·13-s − 0.774·15-s + (−0.363 + 0.630i)17-s + (0.114 + 0.198i)19-s + (0.109 + 0.566i)21-s + (0.312 + 0.541i)23-s + (−0.400 + 0.692i)25-s − 0.962·27-s + 1.11·29-s + (−0.628 + 1.08i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28374 + 0.853926i\)
\(L(\frac12)\) \(\approx\) \(1.28374 + 0.853926i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 1.73i)T \)
good3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−10.91497130605560758270053517083, −10.55280952693750481339003528760, −9.800951027846992821465819306583, −8.555768421115455429291677381838, −7.45512965402255389482431017467, −6.63883669772389404854429973163, −5.55061550291725934902527566818, −4.51220536283492186322563749701, −3.29639997038759109601918355873, −1.80943453809151851296153607180, 1.15276541747503808330505574481, 2.31505432030478399387630814452, 4.42571726722066196781986427174, 5.12460747268659715152536099635, 6.18609951414065760274030415237, 7.17933708136547276787702128735, 8.276656585814744164357637659040, 9.300819492913114517440680641305, 9.649298985050331042242768322384, 11.15332696887654373056707669409

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