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Properties

Label 2-448-112.27-c1-0-5
Degree $2$
Conductor $448$
Sign $0.877 - 0.479i$
Analytic cond. $3.57729$
Root an. cond. $1.89137$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.23 + 2.23i)3-s + (0.584 − 0.584i)5-s + (1.82 − 1.91i)7-s − 6.98i·9-s + (2 − 2i)11-s + (2.91 + 2.91i)13-s + 2.61i·15-s − 2.66i·17-s + (−0.319 + 0.319i)19-s + (0.198 + 8.35i)21-s + 3.27·23-s + 4.31i·25-s + (8.90 + 8.90i)27-s + (−2.04 + 2.04i)29-s − 2.52·31-s + ⋯
L(s)  = 1  + (−1.29 + 1.29i)3-s + (0.261 − 0.261i)5-s + (0.690 − 0.723i)7-s − 2.32i·9-s + (0.603 − 0.603i)11-s + (0.807 + 0.807i)13-s + 0.673i·15-s − 0.645i·17-s + (−0.0733 + 0.0733i)19-s + (0.0432 + 1.82i)21-s + 0.683·23-s + 0.863i·25-s + (1.71 + 1.71i)27-s + (−0.379 + 0.379i)29-s − 0.453·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.877 - 0.479i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.877 - 0.479i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04632 + 0.267187i\)
\(L(\frac12)\) \(\approx\) \(1.04632 + 0.267187i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.82 + 1.91i)T \)
good3 \( 1 + (2.23 - 2.23i)T - 3iT^{2} \)
5 \( 1 + (-0.584 + 0.584i)T - 5iT^{2} \)
11 \( 1 + (-2 + 2i)T - 11iT^{2} \)
13 \( 1 + (-2.91 - 2.91i)T + 13iT^{2} \)
17 \( 1 + 2.66iT - 17T^{2} \)
19 \( 1 + (0.319 - 0.319i)T - 19iT^{2} \)
23 \( 1 - 3.27T + 23T^{2} \)
29 \( 1 + (2.04 - 2.04i)T - 29iT^{2} \)
31 \( 1 + 2.52T + 31T^{2} \)
37 \( 1 + (-1.70 - 1.70i)T + 37iT^{2} \)
41 \( 1 - 11.9T + 41T^{2} \)
43 \( 1 + (-3.27 + 3.27i)T - 43iT^{2} \)
47 \( 1 - 9.96T + 47T^{2} \)
53 \( 1 + (2.37 + 2.37i)T + 53iT^{2} \)
59 \( 1 + (4.14 + 4.14i)T + 59iT^{2} \)
61 \( 1 + (4.52 + 4.52i)T + 61iT^{2} \)
67 \( 1 + (-4.37 - 4.37i)T + 67iT^{2} \)
71 \( 1 + 5.14T + 71T^{2} \)
73 \( 1 + 6.99T + 73T^{2} \)
79 \( 1 - 11.2iT - 79T^{2} \)
83 \( 1 + (-5.39 + 5.39i)T - 83iT^{2} \)
89 \( 1 + 1.05T + 89T^{2} \)
97 \( 1 + 13.2iT - 97T^{2} \)
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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

−11.21464723382686436569581090340, −10.52996296420731146109053906238, −9.381462397449774572769293561097, −8.917374393365774123220433617030, −7.28431072819060485901910960947, −6.20413099157681589646450906888, −5.36119976179975481734639962655, −4.43268547651481900535825099551, −3.66509442176143800464894213006, −1.06650654246536125367966940689, 1.20024364246748373416425444053, 2.37705275802625668385954330958, 4.49329495451580735193977944968, 5.79553451214416603003016243245, 6.07509228949720337302822897894, 7.25901124671968314038624350868, 8.007187696368482929848156680529, 9.127641840588047927528425913271, 10.63996963225675392019217944193, 11.04827741221209052874568326837

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