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Properties

Label 1-448-448.11-r1-0-0
Degree $1$
Conductor $448$
Sign $-0.974 + 0.225i$
Analytic cond. $48.1442$
Root an. cond. $48.1442$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.793 + 0.608i)3-s + (−0.608 − 0.793i)5-s + (0.258 + 0.965i)9-s + (−0.130 − 0.991i)11-s + (−0.382 − 0.923i)13-s − i·15-s + (−0.866 + 0.5i)17-s + (0.991 + 0.130i)19-s + (0.258 + 0.965i)23-s + (−0.258 + 0.965i)25-s + (−0.382 + 0.923i)27-s + (−0.923 + 0.382i)29-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)33-s + (0.608 + 0.793i)37-s + ⋯
L(s)  = 1  + (0.793 + 0.608i)3-s + (−0.608 − 0.793i)5-s + (0.258 + 0.965i)9-s + (−0.130 − 0.991i)11-s + (−0.382 − 0.923i)13-s − i·15-s + (−0.866 + 0.5i)17-s + (0.991 + 0.130i)19-s + (0.258 + 0.965i)23-s + (−0.258 + 0.965i)25-s + (−0.382 + 0.923i)27-s + (−0.923 + 0.382i)29-s + (−0.5 − 0.866i)31-s + (0.5 − 0.866i)33-s + (0.608 + 0.793i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.974 + 0.225i$
Analytic conductor: \(48.1442\)
Root analytic conductor: \(48.1442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 448,\ (1:\ ),\ -0.974 + 0.225i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03475634067 + 0.3049074328i\)
\(L(\frac12)\) \(\approx\) \(0.03475634067 + 0.3049074328i\)
\(L(1)\) \(\approx\) \(0.9731554966 + 0.07809752118i\)
\(L(1)\) \(\approx\) \(0.9731554966 + 0.07809752118i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.793 + 0.608i)T \)
5 \( 1 + (-0.608 - 0.793i)T \)
11 \( 1 + (-0.130 - 0.991i)T \)
13 \( 1 + (-0.382 - 0.923i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.991 + 0.130i)T \)
23 \( 1 + (0.258 + 0.965i)T \)
29 \( 1 + (-0.923 + 0.382i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.608 + 0.793i)T \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 + (-0.923 - 0.382i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (0.130 + 0.991i)T \)
59 \( 1 + (-0.991 + 0.130i)T \)
61 \( 1 + (-0.130 + 0.991i)T \)
67 \( 1 + (0.793 + 0.608i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (-0.965 - 0.258i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (-0.382 - 0.923i)T \)
89 \( 1 + (0.965 - 0.258i)T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.42389674423380607821369315794, −22.67348160101241182634107367943, −21.749708188271492279419623198103, −20.50412724186597576030103622067, −19.97764949885596911660583431713, −19.09119881644300648959124403488, −18.33814526128454608196153188150, −17.72292787726764559201157526264, −16.30214521317891222875679236390, −15.330435424593660543903890398982, −14.6289904725005580465271279397, −13.93473284037072047008103145378, −12.85895587609333945932398827375, −11.9811629015561722223310855399, −11.14915486678470338915519767660, −9.850663674264121766649232910700, −9.06794023637020465517471588369, −7.89951477218240233284305386218, −7.08827552665980217411443265193, −6.60932300644672728625653788097, −4.81545177582244077618376680899, −3.76632851756202553310860055045, −2.687010300975251957015495310741, −1.82201144534934561384763956387, −0.068785606662983916984052174489, 1.45223879554419200844098703297, 3.00515395667868989678696928753, 3.73695216430093637769041852777, 4.86378427000631663242665587404, 5.68902446844682845867176973436, 7.44017543560949372608306286915, 8.13644179256204666367337437678, 8.93534921864135370177936035057, 9.78828371299668082857738950862, 10.9183858927078903982828281329, 11.748631295843149413834074493560, 13.15286392988977032506081432724, 13.462714948688898304482600135016, 14.90856300623057187897969059708, 15.40069660189790848720403903108, 16.32871061145071917905559143734, 16.97707748397390343707243540031, 18.327924817967606855475978336870, 19.317476609489461842867363692226, 20.08998661338168007357715188767, 20.49711195510083193521752121608, 21.66751131054358498495864760782, 22.21066889282572202527124634164, 23.46406117086787221272598176141, 24.40731636996071808378778271975

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