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Properties

Label 2-448-112.19-c1-0-10
Degree $2$
Conductor $448$
Sign $0.486 + 0.873i$
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  + (0.388 − 1.44i)3-s + (2.81 − 0.755i)5-s + (1.47 − 2.19i)7-s + (0.649 + 0.375i)9-s + (−0.0165 + 0.0619i)11-s + (−3.62 + 3.62i)13-s − 4.37i·15-s + (3.26 − 1.88i)17-s + (−2.13 + 0.572i)19-s + (−2.60 − 2.99i)21-s + (−3.80 + 6.59i)23-s + (3.04 − 1.75i)25-s + (3.97 − 3.97i)27-s + (−4.74 − 4.74i)29-s + (0.329 + 0.570i)31-s + ⋯
L(s)  = 1  + (0.224 − 0.836i)3-s + (1.26 − 0.337i)5-s + (0.558 − 0.829i)7-s + (0.216 + 0.125i)9-s + (−0.00500 + 0.0186i)11-s + (−1.00 + 1.00i)13-s − 1.12i·15-s + (0.790 − 0.456i)17-s + (−0.490 + 0.131i)19-s + (−0.568 − 0.653i)21-s + (−0.793 + 1.37i)23-s + (0.608 − 0.351i)25-s + (0.765 − 0.765i)27-s + (−0.880 − 0.880i)29-s + (0.0591 + 0.102i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.486 + 0.873i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.486 + 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64561 - 0.966665i\)
\(L(\frac12)\) \(\approx\) \(1.64561 - 0.966665i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.47 + 2.19i)T \)
good3 \( 1 + (-0.388 + 1.44i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (-2.81 + 0.755i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.0165 - 0.0619i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (3.62 - 3.62i)T - 13iT^{2} \)
17 \( 1 + (-3.26 + 1.88i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.13 - 0.572i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.80 - 6.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.74 + 4.74i)T + 29iT^{2} \)
31 \( 1 + (-0.329 - 0.570i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.08 + 4.05i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 7.67T + 41T^{2} \)
43 \( 1 + (-2.54 - 2.54i)T + 43iT^{2} \)
47 \( 1 + (2.62 - 4.55i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.1 + 2.72i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.43 + 0.652i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.84 - 6.89i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (5.98 + 1.60i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 1.08T + 71T^{2} \)
73 \( 1 + (0.0232 + 0.0402i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-15.2 - 8.80i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.07 - 5.07i)T + 83iT^{2} \)
89 \( 1 + (4.47 - 7.75i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.85iT - 97T^{2} \)
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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

−10.93014058891306765955281170460, −9.771342767736886694462249588162, −9.439323765524127767706072751714, −7.892513713141166187534609324218, −7.41083414936236200860084003534, −6.34297823591747456479816631458, −5.27101841929411522604290219142, −4.17344803731536265950517785781, −2.24396323938358720260189603562, −1.44116291603233712676664213710, 1.99250972292863291790586725309, 3.09103936536382977275518791764, 4.60951470310681683213160848518, 5.49519210783107814727457563999, 6.33797129393659660750219714636, 7.71753960744078133504541915878, 8.763739798634114832109178205054, 9.624357804587167066909121597964, 10.22781402827050933289413307723, 10.87292901868003626916821075153

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