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Properties

Label 2-448-112.19-c1-0-5
Degree $2$
Conductor $448$
Sign $0.809 + 0.587i$
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  + (0.204 − 0.763i)3-s + (−3.81 + 1.02i)5-s + (2.64 + 0.00379i)7-s + (2.05 + 1.18i)9-s + (1.16 − 4.35i)11-s + (1.34 − 1.34i)13-s + 3.12i·15-s + (0.989 − 0.571i)17-s + (3.01 − 0.808i)19-s + (0.544 − 2.01i)21-s + (1.02 − 1.77i)23-s + (9.16 − 5.29i)25-s + (3.00 − 3.00i)27-s + (5.30 + 5.30i)29-s + (−2.07 − 3.58i)31-s + ⋯
L(s)  = 1  + (0.118 − 0.440i)3-s + (−1.70 + 0.456i)5-s + (0.999 + 0.00143i)7-s + (0.685 + 0.395i)9-s + (0.352 − 1.31i)11-s + (0.374 − 0.374i)13-s + 0.805i·15-s + (0.239 − 0.138i)17-s + (0.691 − 0.185i)19-s + (0.118 − 0.440i)21-s + (0.213 − 0.369i)23-s + (1.83 − 1.05i)25-s + (0.578 − 0.578i)27-s + (0.984 + 0.984i)29-s + (−0.372 − 0.644i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 + 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.809 + 0.587i$
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: \((2,\ 448,\ (\ :1/2),\ 0.809 + 0.587i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24814 - 0.405406i\)
\(L(\frac12)\) \(\approx\) \(1.24814 - 0.405406i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 - 0.00379i)T \)
good3 \( 1 + (-0.204 + 0.763i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (3.81 - 1.02i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.16 + 4.35i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.34 + 1.34i)T - 13iT^{2} \)
17 \( 1 + (-0.989 + 0.571i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.01 + 0.808i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.02 + 1.77i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.30 - 5.30i)T + 29iT^{2} \)
31 \( 1 + (2.07 + 3.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.803 + 2.99i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 8.08T + 41T^{2} \)
43 \( 1 + (-3.82 - 3.82i)T + 43iT^{2} \)
47 \( 1 + (0.814 - 1.41i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.92 + 0.783i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (6.97 + 1.86i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.68 - 6.30i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.616 + 0.165i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 2.40T + 71T^{2} \)
73 \( 1 + (-1.47 - 2.56i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.06 + 0.613i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.29 - 2.29i)T + 83iT^{2} \)
89 \( 1 + (-4.36 + 7.56i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.42iT - 97T^{2} \)
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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

−11.11537177764721858749891927326, −10.50845853568859775881605833381, −8.826802508695548712757981986659, −8.064523832282241053141025936777, −7.56056838473916140012590274985, −6.60621027838475200592275478338, −5.11193334016687375001888166970, −4.03357319126755673210282662027, −3.02337208695190037840028695814, −1.03890258870842151369923757113, 1.39343307911068813172692170009, 3.56298596137885075639904486487, 4.37198443392840598497721963753, 4.96928874231579145332340815684, 6.86320417976499589676295680590, 7.60288040071196975921321338051, 8.390723036010383807716281454554, 9.318673749428354862851937341665, 10.32031779585962516558865561871, 11.35843197220945043019187799708

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