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Properties

Label 2-448-112.37-c1-0-10
Degree $2$
Conductor $448$
Sign $-0.496 + 0.868i$
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 − 0.133i)3-s + (−0.866 + 0.232i)5-s + (−1.73 + 2i)7-s + (−2.36 − 1.36i)9-s + (0.767 − 2.86i)11-s + (3.73 − 3.73i)13-s + 0.464·15-s + (−3.23 − 5.59i)17-s + (−0.767 − 2.86i)19-s + (1.13 − 0.767i)21-s + (−3.86 − 2.23i)23-s + (−3.63 + 2.09i)25-s + (2.09 + 2.09i)27-s + (−0.267 + 0.267i)29-s + (1.86 + 3.23i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.0773i)3-s + (−0.387 + 0.103i)5-s + (−0.654 + 0.755i)7-s + (−0.788 − 0.455i)9-s + (0.231 − 0.864i)11-s + (1.03 − 1.03i)13-s + 0.119·15-s + (−0.783 − 1.35i)17-s + (−0.176 − 0.657i)19-s + (0.247 − 0.167i)21-s + (−0.806 − 0.465i)23-s + (−0.726 + 0.419i)25-s + (0.403 + 0.403i)27-s + (−0.0497 + 0.0497i)29-s + (0.335 + 0.580i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.496 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.496 + 0.868i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ -0.496 + 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.300205 - 0.517489i\)
\(L(\frac12)\) \(\approx\) \(0.300205 - 0.517489i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.133i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (0.866 - 0.232i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.767 + 2.86i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.73 + 3.73i)T - 13iT^{2} \)
17 \( 1 + (3.23 + 5.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.767 + 2.86i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.86 + 2.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.267 - 0.267i)T - 29iT^{2} \)
31 \( 1 + (-1.86 - 3.23i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.13 - 0.303i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 4.92iT - 41T^{2} \)
43 \( 1 + (6.46 + 6.46i)T + 43iT^{2} \)
47 \( 1 + (-2.13 + 3.69i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.06 - 3.96i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.03 + 11.3i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.86 - 6.96i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-4.96 - 1.33i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 0.535iT - 71T^{2} \)
73 \( 1 + (-6.23 + 3.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.33 - 14.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.53 - 1.53i)T - 83iT^{2} \)
89 \( 1 + (4.5 + 2.59i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.92T + 97T^{2} \)
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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

−11.11475904959684812894517795057, −9.869816197566016799585536569103, −8.800762254127290577885859091601, −8.357156592722837074529701682864, −6.86859940229858690171282895707, −6.07710796611901039015105978042, −5.26881243596349427223234032556, −3.61588784724454165561660084360, −2.76220003124366844471508425636, −0.37453652361605750937777063427, 1.89025449996473466413270251347, 3.76438537951813727616763377487, 4.35821979345293083055690914896, 5.96219296656582741867526358105, 6.58717842445407537751557956472, 7.80941568450403754568900279315, 8.611336664561850667853607198688, 9.723262190373063684642897750492, 10.55193227063121937087048605763, 11.37133117230138458430461840074

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