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Properties

Label 2-448-112.27-c1-0-1
Degree $2$
Conductor $448$
Sign $-0.710 - 0.703i$
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  + (−2.44 + 2.44i)5-s + (1 − 2.44i)7-s + 3i·9-s + (−1 + i)11-s + (−2.44 − 2.44i)13-s + 4.89i·17-s + (−4.89 + 4.89i)19-s − 4·23-s − 6.99i·25-s + (−3 + 3i)29-s − 4.89·31-s + (3.55 + 8.44i)35-s + (5 + 5i)37-s + 4.89·41-s + (5 − 5i)43-s + ⋯
L(s)  = 1  + (−1.09 + 1.09i)5-s + (0.377 − 0.925i)7-s + i·9-s + (−0.301 + 0.301i)11-s + (−0.679 − 0.679i)13-s + 1.18i·17-s + (−1.12 + 1.12i)19-s − 0.834·23-s − 1.39i·25-s + (−0.557 + 0.557i)29-s − 0.879·31-s + (0.600 + 1.42i)35-s + (0.821 + 0.821i)37-s + 0.765·41-s + (0.762 − 0.762i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.710 - 0.703i$
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: \((2,\ 448,\ (\ :1/2),\ -0.710 - 0.703i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.250569 + 0.609319i\)
\(L(\frac12)\) \(\approx\) \(0.250569 + 0.609319i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 2.44i)T \)
good3 \( 1 - 3iT^{2} \)
5 \( 1 + (2.44 - 2.44i)T - 5iT^{2} \)
11 \( 1 + (1 - i)T - 11iT^{2} \)
13 \( 1 + (2.44 + 2.44i)T + 13iT^{2} \)
17 \( 1 - 4.89iT - 17T^{2} \)
19 \( 1 + (4.89 - 4.89i)T - 19iT^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + (3 - 3i)T - 29iT^{2} \)
31 \( 1 + 4.89T + 31T^{2} \)
37 \( 1 + (-5 - 5i)T + 37iT^{2} \)
41 \( 1 - 4.89T + 41T^{2} \)
43 \( 1 + (-5 + 5i)T - 43iT^{2} \)
47 \( 1 - 4.89T + 47T^{2} \)
53 \( 1 + (1 + i)T + 53iT^{2} \)
59 \( 1 + (-4.89 - 4.89i)T + 59iT^{2} \)
61 \( 1 + (-2.44 - 2.44i)T + 61iT^{2} \)
67 \( 1 + (5 + 5i)T + 67iT^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 9.79T + 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 4.89iT - 97T^{2} \)
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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

−11.10413863478374454831067611661, −10.61839176702092757186078008986, −10.11831261469570580711498185561, −8.268148703722332397247435283334, −7.74943421172803626168589347583, −7.13589588782950876192318367053, −5.82806740716417429767797196068, −4.41082396461164857336095671293, −3.65168361519705358505310034914, −2.16610385205557806691511052835, 0.39574006986261279509391240487, 2.44230968382041748701932054782, 4.03973201023045790626887351144, 4.81293380338181100673721994054, 5.91903418261671253179426776646, 7.21628753749042613334592326635, 8.122338433040946297856386497583, 9.094963027553661249744154547068, 9.373919491995049782927842159920, 11.16206859475083615479126115834

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