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Properties

Label 2-448-112.109-c1-0-9
Degree $2$
Conductor $448$
Sign $0.899 + 0.437i$
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  + (3.04 − 0.814i)3-s + (−1.87 − 0.501i)5-s + (1.89 + 1.84i)7-s + (5.98 − 3.45i)9-s + (−0.299 − 1.11i)11-s + (0.00680 + 0.00680i)13-s − 6.10·15-s + (1.52 − 2.63i)17-s + (−1.24 + 4.64i)19-s + (7.27 + 4.05i)21-s + (4.27 − 2.46i)23-s + (−1.07 − 0.620i)25-s + (8.70 − 8.70i)27-s + (−1.45 − 1.45i)29-s + (−2.60 + 4.51i)31-s + ⋯
L(s)  = 1  + (1.75 − 0.470i)3-s + (−0.837 − 0.224i)5-s + (0.717 + 0.696i)7-s + (1.99 − 1.15i)9-s + (−0.0904 − 0.337i)11-s + (0.00188 + 0.00188i)13-s − 1.57·15-s + (0.369 − 0.639i)17-s + (−0.285 + 1.06i)19-s + (1.58 + 0.885i)21-s + (0.891 − 0.514i)23-s + (−0.214 − 0.124i)25-s + (1.67 − 1.67i)27-s + (−0.270 − 0.270i)29-s + (−0.468 + 0.811i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.899 + 0.437i$
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: \((2,\ 448,\ (\ :1/2),\ 0.899 + 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.24131 - 0.515865i\)
\(L(\frac12)\) \(\approx\) \(2.24131 - 0.515865i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.89 - 1.84i)T \)
good3 \( 1 + (-3.04 + 0.814i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (1.87 + 0.501i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.299 + 1.11i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-0.00680 - 0.00680i)T + 13iT^{2} \)
17 \( 1 + (-1.52 + 2.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.24 - 4.64i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-4.27 + 2.46i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.45 + 1.45i)T + 29iT^{2} \)
31 \( 1 + (2.60 - 4.51i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.51 + 2.28i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 6.02iT - 41T^{2} \)
43 \( 1 + (7.17 - 7.17i)T - 43iT^{2} \)
47 \( 1 + (0.796 + 1.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.211 - 0.787i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.94 + 7.25i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.46 + 9.21i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (6.35 - 1.70i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 6.77iT - 71T^{2} \)
73 \( 1 + (-3.43 - 1.98i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.81 - 4.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.3 - 11.3i)T + 83iT^{2} \)
89 \( 1 + (3.59 - 2.07i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.390T + 97T^{2} \)
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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

−11.13824514937263132894363831108, −9.827647656164683509103728521829, −8.908495691018604529670353413259, −8.217553061336960711770271358698, −7.81352653069628428531833261093, −6.71112856642384457085728775191, −5.08444115580718246360541307339, −3.83548450154784048759993172900, −2.90323420552293654864399361193, −1.63030634534452990899188597291, 1.88977193047568663508127765222, 3.31572943911225651534407238345, 4.00575284572855491466251375769, 4.98809479320231650097439354429, 7.15339481958627686809144299733, 7.55405796888908049776469793306, 8.477440688712784140031037186662, 9.138862729679573249921088226405, 10.25608255948070625185263577584, 10.89958039662295909462428368562

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