Properties

Label 2-448-16.3-c4-0-38
Degree $2$
Conductor $448$
Sign $0.328 + 0.944i$
Analytic cond. $46.3097$
Root an. cond. $6.80512$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.91 + 1.91i)3-s + (33.8 − 33.8i)5-s − 18.5·7-s + 73.6i·9-s + (−100. − 100. i)11-s + (28.6 + 28.6i)13-s + 129. i·15-s + 549.·17-s + (−58.0 + 58.0i)19-s + (35.3 − 35.3i)21-s + 874.·23-s − 1.66e3i·25-s + (−295. − 295. i)27-s + (−307. − 307. i)29-s + 763. i·31-s + ⋯
L(s)  = 1  + (−0.212 + 0.212i)3-s + (1.35 − 1.35i)5-s − 0.377·7-s + 0.909i·9-s + (−0.832 − 0.832i)11-s + (0.169 + 0.169i)13-s + 0.574i·15-s + 1.90·17-s + (−0.160 + 0.160i)19-s + (0.0802 − 0.0802i)21-s + 1.65·23-s − 2.66i·25-s + (−0.405 − 0.405i)27-s + (−0.365 − 0.365i)29-s + 0.794i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.328 + 0.944i$
Analytic conductor: \(46.3097\)
Root analytic conductor: \(6.80512\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :2),\ 0.328 + 0.944i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.267586792\)
\(L(\frac12)\) \(\approx\) \(2.267586792\)
\(L(3)\) 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 \( 1 + 18.5T \)
good3 \( 1 + (1.91 - 1.91i)T - 81iT^{2} \)
5 \( 1 + (-33.8 + 33.8i)T - 625iT^{2} \)
11 \( 1 + (100. + 100. i)T + 1.46e4iT^{2} \)
13 \( 1 + (-28.6 - 28.6i)T + 2.85e4iT^{2} \)
17 \( 1 - 549.T + 8.35e4T^{2} \)
19 \( 1 + (58.0 - 58.0i)T - 1.30e5iT^{2} \)
23 \( 1 - 874.T + 2.79e5T^{2} \)
29 \( 1 + (307. + 307. i)T + 7.07e5iT^{2} \)
31 \( 1 - 763. iT - 9.23e5T^{2} \)
37 \( 1 + (292. - 292. i)T - 1.87e6iT^{2} \)
41 \( 1 + 1.39e3iT - 2.82e6T^{2} \)
43 \( 1 + (757. + 757. i)T + 3.41e6iT^{2} \)
47 \( 1 - 494. iT - 4.87e6T^{2} \)
53 \( 1 + (-203. + 203. i)T - 7.89e6iT^{2} \)
59 \( 1 + (3.35e3 + 3.35e3i)T + 1.21e7iT^{2} \)
61 \( 1 + (4.49e3 + 4.49e3i)T + 1.38e7iT^{2} \)
67 \( 1 + (-5.20e3 + 5.20e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 6.52e3T + 2.54e7T^{2} \)
73 \( 1 - 1.86e3iT - 2.83e7T^{2} \)
79 \( 1 + 7.14e3iT - 3.89e7T^{2} \)
83 \( 1 + (-1.75e3 + 1.75e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 1.37e3iT - 6.27e7T^{2} \)
97 \( 1 - 5.95e3T + 8.85e7T^{2} \)
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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

−10.22900609723392854513964427355, −9.468305246535995031364572339277, −8.604753636730349094300725700522, −7.76021824678489557125080110368, −6.21704842184483116267588647163, −5.31896109982999100838619646679, −4.99655839069130911195837283193, −3.21450227605890246194441089183, −1.83373013581934680455876023321, −0.67914906067172452972943582401, 1.23814063880980757424529511288, 2.63759826017016139440827795822, 3.38323221789078265376555405631, 5.28089005580843538379080030423, 6.01994096390805390843107273276, 6.88974355435753896711144917122, 7.56523218126487514778166357009, 9.220100701753227515929615818525, 9.879548076740442460654877949640, 10.46614721610612122382581735332

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