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Properties

Label 2-448-28.19-c3-0-4
Degree $2$
Conductor $448$
Sign $0.957 - 0.288i$
Analytic cond. $26.4328$
Root an. cond. $5.14128$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.65 − 8.06i)3-s + (−5.11 − 2.95i)5-s + (1.92 − 18.4i)7-s + (−29.8 + 51.6i)9-s + (−0.267 + 0.154i)11-s + 43.7i·13-s + 54.9i·15-s + (−27.0 + 15.6i)17-s + (−39.7 + 68.8i)19-s + (−157. + 70.1i)21-s + (16.5 + 9.52i)23-s + (−45.0 − 78.0i)25-s + 303.·27-s + 40.1·29-s + (42.8 + 74.2i)31-s + ⋯
L(s)  = 1  + (−0.895 − 1.55i)3-s + (−0.457 − 0.264i)5-s + (0.104 − 0.994i)7-s + (−1.10 + 1.91i)9-s + (−0.00733 + 0.00423i)11-s + 0.932i·13-s + 0.946i·15-s + (−0.386 + 0.223i)17-s + (−0.479 + 0.830i)19-s + (−1.63 + 0.729i)21-s + (0.149 + 0.0863i)23-s + (−0.360 − 0.624i)25-s + 2.16·27-s + 0.256·29-s + (0.248 + 0.430i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.957 - 0.288i$
Analytic conductor: \(26.4328\)
Root analytic conductor: \(5.14128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :3/2),\ 0.957 - 0.288i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5075453434\)
\(L(\frac12)\) \(\approx\) \(0.5075453434\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.92 + 18.4i)T \)
good3 \( 1 + (4.65 + 8.06i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (5.11 + 2.95i)T + (62.5 + 108. i)T^{2} \)
11 \( 1 + (0.267 - 0.154i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 43.7iT - 2.19e3T^{2} \)
17 \( 1 + (27.0 - 15.6i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (39.7 - 68.8i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-16.5 - 9.52i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 40.1T + 2.43e4T^{2} \)
31 \( 1 + (-42.8 - 74.2i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (72.4 - 125. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 254. iT - 6.89e4T^{2} \)
43 \( 1 + 366. iT - 7.95e4T^{2} \)
47 \( 1 + (5.12 - 8.87i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (315. + 546. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-257. - 446. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-669. - 386. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-450. + 259. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 261. iT - 3.57e5T^{2} \)
73 \( 1 + (-588. + 339. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (299. + 172. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 805.T + 5.71e5T^{2} \)
89 \( 1 + (40.7 + 23.5i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 763. iT - 9.12e5T^{2} \)
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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

−11.00545914416135307063992674808, −10.08475361689581260998907879523, −8.497368836973818088561414647564, −7.83507770299879752825987932075, −6.86696028946125397314318041808, −6.39245144402913579663009021840, −5.08092109632802843967417934730, −3.96120629388547398068858634959, −2.02790136213562454620582986870, −0.961022829921070248312949009224, 0.23166319796105913900465715351, 2.76145151197298689353952576819, 3.88025043670143244986744078803, 4.94480832511804792488538348236, 5.61129042277035784900869875611, 6.63031632772264630677344016990, 8.109312308197776064308359217633, 9.112540511550688701343816148719, 9.770441864644129863577642845938, 10.87412670442892012440955780504

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