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Properties

Label 2-448-28.3-c3-0-1
Degree $2$
Conductor $448$
Sign $-0.741 + 0.670i$
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  + (−2.11 + 3.65i)3-s + (−1.03 + 0.596i)5-s + (−16.7 − 7.86i)7-s + (4.58 + 7.93i)9-s + (29.6 + 17.0i)11-s + 56.1i·13-s − 5.04i·15-s + (19.4 + 11.2i)17-s + (−53.4 − 92.4i)19-s + (64.1 − 44.7i)21-s + (51.2 − 29.5i)23-s + (−61.7 + 107. i)25-s − 152.·27-s − 211.·29-s + (−54.1 + 93.8i)31-s + ⋯
L(s)  = 1  + (−0.406 + 0.703i)3-s + (−0.0924 + 0.0533i)5-s + (−0.905 − 0.424i)7-s + (0.169 + 0.293i)9-s + (0.811 + 0.468i)11-s + 1.19i·13-s − 0.0867i·15-s + (0.277 + 0.160i)17-s + (−0.644 − 1.11i)19-s + (0.666 − 0.464i)21-s + (0.464 − 0.268i)23-s + (−0.494 + 0.856i)25-s − 1.08·27-s − 1.35·29-s + (−0.313 + 0.543i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 + 0.670i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.741 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.741 + 0.670i$
Analytic conductor: \(26.4328\)
Root analytic conductor: \(5.14128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :3/2),\ -0.741 + 0.670i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1486694896\)
\(L(\frac12)\) \(\approx\) \(0.1486694896\)
\(L(\frac{5}{2})\) 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 \( 1 + (16.7 + 7.86i)T \)
good3 \( 1 + (2.11 - 3.65i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (1.03 - 0.596i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (-29.6 - 17.0i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 56.1iT - 2.19e3T^{2} \)
17 \( 1 + (-19.4 - 11.2i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (53.4 + 92.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-51.2 + 29.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 211.T + 2.43e4T^{2} \)
31 \( 1 + (54.1 - 93.8i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-81.0 - 140. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 414. iT - 6.89e4T^{2} \)
43 \( 1 + 258. iT - 7.95e4T^{2} \)
47 \( 1 + (304. + 527. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-112. + 194. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (42.9 - 74.3i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-312. + 180. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (580. + 334. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 133. iT - 3.57e5T^{2} \)
73 \( 1 + (675. + 389. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (729. - 421. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 432.T + 5.71e5T^{2} \)
89 \( 1 + (439. - 253. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 372. iT - 9.12e5T^{2} \)
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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

−11.15097048329308923367956049188, −10.32588003031739528776961904172, −9.490146890666041607090906004994, −8.879010451679481685531077940483, −7.22624421103913902504719870448, −6.75707928528048709868785036893, −5.45431105985059104926492657697, −4.34287624415563443325160332122, −3.63226951008336159217483876908, −1.88095176599700458845698340664, 0.05224373808412282746320877627, 1.33975183567445187734617062046, 2.99747934831582945420380056995, 4.05721059111217472977963891543, 5.90544418046571466731642275756, 6.05963952700375474127360124673, 7.29301595268208711905337525504, 8.179620623650142185418014061822, 9.335844193382109483752795178480, 10.00980092585058624664246991576

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