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Properties

Label 2-448-7.6-c2-0-5
Degree $2$
Conductor $448$
Sign $-0.887 + 0.461i$
Analytic cond. $12.2071$
Root an. cond. $3.49386$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.43i·3-s + 6.12i·5-s + (6.21 − 3.23i)7-s − 20.5·9-s − 15.2·11-s + 3.00i·13-s − 33.3·15-s + 21.6i·17-s − 11.8i·19-s + (17.5 + 33.7i)21-s + 1.72·23-s − 12.5·25-s − 62.8i·27-s + 41.1·29-s − 8.50i·31-s + ⋯
L(s)  = 1  + 1.81i·3-s + 1.22i·5-s + (0.887 − 0.461i)7-s − 2.28·9-s − 1.38·11-s + 0.231i·13-s − 2.22·15-s + 1.27i·17-s − 0.626i·19-s + (0.836 + 1.60i)21-s + 0.0748·23-s − 0.502·25-s − 2.32i·27-s + 1.41·29-s − 0.274i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.887 + 0.461i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ -0.887 + 0.461i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.228349911\)
\(L(\frac12)\) \(\approx\) \(1.228349911\)
\(L(2)\) 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 \( 1 + (-6.21 + 3.23i)T \)
good3 \( 1 - 5.43iT - 9T^{2} \)
5 \( 1 - 6.12iT - 25T^{2} \)
11 \( 1 + 15.2T + 121T^{2} \)
13 \( 1 - 3.00iT - 169T^{2} \)
17 \( 1 - 21.6iT - 289T^{2} \)
19 \( 1 + 11.8iT - 361T^{2} \)
23 \( 1 - 1.72T + 529T^{2} \)
29 \( 1 - 41.1T + 841T^{2} \)
31 \( 1 + 8.50iT - 961T^{2} \)
37 \( 1 + 53.1T + 1.36e3T^{2} \)
41 \( 1 - 43.0iT - 1.68e3T^{2} \)
43 \( 1 + 36.6T + 1.84e3T^{2} \)
47 \( 1 + 30.2iT - 2.20e3T^{2} \)
53 \( 1 - 30T + 2.80e3T^{2} \)
59 \( 1 - 73.0iT - 3.48e3T^{2} \)
61 \( 1 + 5.67iT - 3.72e3T^{2} \)
67 \( 1 - 28.7T + 4.48e3T^{2} \)
71 \( 1 - 5.53T + 5.04e3T^{2} \)
73 \( 1 - 94.9iT - 5.32e3T^{2} \)
79 \( 1 - 81.5T + 6.24e3T^{2} \)
83 \( 1 + 86.2iT - 6.88e3T^{2} \)
89 \( 1 + 27.6iT - 7.92e3T^{2} \)
97 \( 1 - 100. iT - 9.40e3T^{2} \)
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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

−10.94881851550043141572107449279, −10.44190176099591493997657887626, −10.11833417428346855704614713248, −8.725346024292644286515680049687, −7.980547251063529967176739231301, −6.69699929787677922216445914668, −5.41296079648523479402855401423, −4.61266215396720601723518585367, −3.58073443980855305138723091982, −2.56279088805458355110054179458, 0.50169817016792925087487846483, 1.67140672544115934542138631581, 2.75180132977316826928167585303, 5.03449204480899484932365337876, 5.42796633788734408403857587772, 6.78165743687652202850429811075, 7.83721587951755282247087036489, 8.239335241373027190998446621954, 9.035854529868835315577946953646, 10.55067582847726102410574845678

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