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Properties

Label 2-448-448.69-c0-0-0
Degree $2$
Conductor $448$
Sign $0.881 - 0.471i$
Analytic cond. $0.223581$
Root an. cond. $0.472843$
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  + (−0.923 − 0.382i)2-s + (0.707 + 0.707i)4-s + (−0.382 + 0.923i)7-s + (−0.382 − 0.923i)8-s + (0.382 + 0.923i)9-s + (0.216 + 0.324i)11-s + (0.707 − 0.707i)14-s + i·16-s − i·18-s + (−0.0761 − 0.382i)22-s + (−0.707 + 0.292i)23-s + (0.923 + 0.382i)25-s + (−0.923 + 0.382i)28-s + (1.08 − 1.63i)29-s + (0.382 − 0.923i)32-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)2-s + (0.707 + 0.707i)4-s + (−0.382 + 0.923i)7-s + (−0.382 − 0.923i)8-s + (0.382 + 0.923i)9-s + (0.216 + 0.324i)11-s + (0.707 − 0.707i)14-s + i·16-s − i·18-s + (−0.0761 − 0.382i)22-s + (−0.707 + 0.292i)23-s + (0.923 + 0.382i)25-s + (−0.923 + 0.382i)28-s + (1.08 − 1.63i)29-s + (0.382 − 0.923i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.881 - 0.471i$
Analytic conductor: \(0.223581\)
Root analytic conductor: \(0.472843\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :0),\ 0.881 - 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5697546341\)
\(L(\frac12)\) \(\approx\) \(0.5697546341\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 + 0.382i)T \)
7 \( 1 + (0.382 - 0.923i)T \)
good3 \( 1 + (-0.382 - 0.923i)T^{2} \)
5 \( 1 + (-0.923 - 0.382i)T^{2} \)
11 \( 1 + (-0.216 - 0.324i)T + (-0.382 + 0.923i)T^{2} \)
13 \( 1 + (-0.923 + 0.382i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (0.923 - 0.382i)T^{2} \)
23 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
29 \( 1 + (-1.08 + 1.63i)T + (-0.382 - 0.923i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.216 - 1.08i)T + (-0.923 - 0.382i)T^{2} \)
41 \( 1 + (-0.707 + 0.707i)T^{2} \)
43 \( 1 + (-0.923 + 0.617i)T + (0.382 - 0.923i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.923 + 1.38i)T + (-0.382 + 0.923i)T^{2} \)
59 \( 1 + (-0.923 - 0.382i)T^{2} \)
61 \( 1 + (-0.382 - 0.923i)T^{2} \)
67 \( 1 + (1.38 + 0.923i)T + (0.382 + 0.923i)T^{2} \)
71 \( 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
83 \( 1 + (0.923 - 0.382i)T^{2} \)
89 \( 1 + (-0.707 - 0.707i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−11.34948047618141224906462467461, −10.27632776197962310114965217581, −9.712525095290340289314030356806, −8.707448530013404193930938567569, −7.966685684332821679095444917333, −6.95417572609097357369801322638, −5.93390352863081467430846390822, −4.51482934039070637162074868623, −2.98859299644020148231431846203, −1.89517604791095561611950382313, 1.12583501653622014575534085598, 3.08755972594004015755799895107, 4.45594372290695617134425682395, 5.99818276824082157217105210353, 6.76143738555051994887725431505, 7.49623544894324317453541514918, 8.665463336954115822841763981658, 9.364999083081969987891016650275, 10.33164679791537747592832051844, 10.84210083170122706390521977863

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