Properties

Label 2-448-28.3-c1-0-13
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  + (1.32 − 2.29i)3-s + (1.5 − 0.866i)5-s − 2.64·7-s + (−2 − 3.46i)9-s + (−3.96 − 2.29i)11-s − 3.46i·13-s − 4.58i·15-s + (4.5 + 2.59i)17-s + (1.32 + 2.29i)19-s + (−3.50 + 6.06i)21-s + (3.96 − 2.29i)23-s + (−1 + 1.73i)25-s − 2.64·27-s + (1.32 − 2.29i)31-s + (−10.5 + 6.06i)33-s + ⋯
L(s)  = 1  + (0.763 − 1.32i)3-s + (0.670 − 0.387i)5-s − 0.999·7-s + (−0.666 − 1.15i)9-s + (−1.19 − 0.690i)11-s − 0.960i·13-s − 1.18i·15-s + (1.09 + 0.630i)17-s + (0.303 + 0.525i)19-s + (−0.763 + 1.32i)21-s + (0.827 − 0.477i)23-s + (−0.200 + 0.346i)25-s − 0.509·27-s + (0.237 − 0.411i)31-s + (−1.82 + 1.05i)33-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: $\chi_{448} (255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.926713 - 1.39317i\)
\(L(\frac12)\) \(\approx\) \(0.926713 - 1.39317i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.64T \)
good3 \( 1 + (-1.32 + 2.29i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.96 + 2.29i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (-4.5 - 2.59i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.32 - 2.29i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.96 + 2.29i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-1.32 + 2.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 + 9.16iT - 43T^{2} \)
47 \( 1 + (-3.96 - 6.87i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.96 - 6.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.5 - 0.866i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.96 - 2.29i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.16iT - 71T^{2} \)
73 \( 1 + (4.5 + 2.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.96 + 2.29i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-1.5 + 0.866i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.46iT - 97T^{2} \)
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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

−10.61228835978616700584541547220, −9.888013900395411405773847271451, −8.796538602939488660827700703048, −8.025061254487974546172989366329, −7.30190410586901942301832239600, −6.07847175169616642711416352293, −5.47556899946297274459248552212, −3.34520832783832062512994153738, −2.55504560494110432051271872280, −0.999125218425913087783433789104, 2.51876891507644918163544429969, 3.27061291014582210909593007890, 4.54157641700522856174806261109, 5.49045783669412762414320363526, 6.76822808675233184757884014788, 7.79863633494701054888935615476, 9.151523105135185430978033530806, 9.624301997540606079359239143350, 10.15054397986786421086910394979, 11.00663386241407087987032415680

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