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Properties

Label 2-448-28.19-c3-0-6
Degree $2$
Conductor $448$
Sign $0.858 - 0.511i$
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  + (−2.69 − 4.66i)3-s + (−12.5 − 7.24i)5-s + (18.3 − 2.59i)7-s + (−1.00 + 1.74i)9-s + (−50.5 + 29.2i)11-s + 35.9i·13-s + 78.0i·15-s + (−47.6 + 27.4i)17-s + (−2.40 + 4.16i)19-s + (−61.5 − 78.5i)21-s + (−22.4 − 12.9i)23-s + (42.3 + 73.3i)25-s − 134.·27-s + 260.·29-s + (−61.1 − 105. i)31-s + ⋯
L(s)  = 1  + (−0.518 − 0.897i)3-s + (−1.12 − 0.647i)5-s + (0.990 − 0.140i)7-s + (−0.0373 + 0.0646i)9-s + (−1.38 + 0.800i)11-s + 0.766i·13-s + 1.34i·15-s + (−0.679 + 0.392i)17-s + (−0.0290 + 0.0503i)19-s + (−0.639 − 0.816i)21-s + (−0.203 − 0.117i)23-s + (0.338 + 0.586i)25-s − 0.959·27-s + 1.67·29-s + (−0.354 − 0.613i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.511i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.858 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.858 - 0.511i$
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: \((2,\ 448,\ (\ :3/2),\ 0.858 - 0.511i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7270830798\)
\(L(\frac12)\) \(\approx\) \(0.7270830798\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.3 + 2.59i)T \)
good3 \( 1 + (2.69 + 4.66i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (12.5 + 7.24i)T + (62.5 + 108. i)T^{2} \)
11 \( 1 + (50.5 - 29.2i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 35.9iT - 2.19e3T^{2} \)
17 \( 1 + (47.6 - 27.4i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (2.40 - 4.16i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (22.4 + 12.9i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 260.T + 2.43e4T^{2} \)
31 \( 1 + (61.1 + 105. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-170. + 294. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 500. iT - 6.89e4T^{2} \)
43 \( 1 - 205. iT - 7.95e4T^{2} \)
47 \( 1 + (96.7 - 167. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-277. - 479. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (99.3 + 172. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-309. - 178. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-779. + 450. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 451. iT - 3.57e5T^{2} \)
73 \( 1 + (-65.0 + 37.5i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (405. + 234. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 624.T + 5.71e5T^{2} \)
89 \( 1 + (525. + 303. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 873. iT - 9.12e5T^{2} \)
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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

−11.17631462880604195915071850536, −9.947539400290984796551874029274, −8.609870630400726532185485006418, −7.84533469665844685483319117901, −7.30949207363823318057308469846, −6.14668805570333565796739903987, −4.79711377787675997089416814294, −4.24997827521555653037766081660, −2.24812061693247070939769648742, −0.964392671320781403178382931599, 0.31810212613023415286637399533, 2.61157140497502903690171934754, 3.78495775834857918678878043970, 4.88481017156787507188102142562, 5.48953773516580043636340369838, 7.02526784757027925556605418935, 8.003329353527342647034613946482, 8.525604856949294796072315903492, 10.17210814675859257938457505719, 10.67433832868454823704435355815

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