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Properties

Label 2-448-112.27-c3-0-1
Degree $2$
Conductor $448$
Sign $-0.960 + 0.278i$
Analytic cond. $26.4328$
Root an. cond. $5.14128$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.04 + 2.04i)3-s + (−1.81 + 1.81i)5-s + (10.2 − 15.4i)7-s + 18.6i·9-s + (−9.40 + 9.40i)11-s + (−31.8 − 31.8i)13-s − 7.41i·15-s + 99.6i·17-s + (105. − 105. i)19-s + (10.6 + 52.3i)21-s − 196.·23-s + 118. i·25-s + (−93.1 − 93.1i)27-s + (−0.691 + 0.691i)29-s + 152.·31-s + ⋯
L(s)  = 1  + (−0.392 + 0.392i)3-s + (−0.162 + 0.162i)5-s + (0.552 − 0.833i)7-s + 0.691i·9-s + (−0.257 + 0.257i)11-s + (−0.679 − 0.679i)13-s − 0.127i·15-s + 1.42i·17-s + (1.27 − 1.27i)19-s + (0.110 + 0.544i)21-s − 1.77·23-s + 0.947i·25-s + (−0.664 − 0.664i)27-s + (−0.00442 + 0.00442i)29-s + 0.886·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.278i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.960 + 0.278i$
Analytic conductor: \(26.4328\)
Root analytic conductor: \(5.14128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :3/2),\ -0.960 + 0.278i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.07981963641\)
\(L(\frac12)\) \(\approx\) \(0.07981963641\)
\(L(\frac{5}{2})\) 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 \( 1 + (-10.2 + 15.4i)T \)
good3 \( 1 + (2.04 - 2.04i)T - 27iT^{2} \)
5 \( 1 + (1.81 - 1.81i)T - 125iT^{2} \)
11 \( 1 + (9.40 - 9.40i)T - 1.33e3iT^{2} \)
13 \( 1 + (31.8 + 31.8i)T + 2.19e3iT^{2} \)
17 \( 1 - 99.6iT - 4.91e3T^{2} \)
19 \( 1 + (-105. + 105. i)T - 6.85e3iT^{2} \)
23 \( 1 + 196.T + 1.21e4T^{2} \)
29 \( 1 + (0.691 - 0.691i)T - 2.43e4iT^{2} \)
31 \( 1 - 152.T + 2.97e4T^{2} \)
37 \( 1 + (-84.9 - 84.9i)T + 5.06e4iT^{2} \)
41 \( 1 + 321.T + 6.89e4T^{2} \)
43 \( 1 + (149. - 149. i)T - 7.95e4iT^{2} \)
47 \( 1 + 422.T + 1.03e5T^{2} \)
53 \( 1 + (449. + 449. i)T + 1.48e5iT^{2} \)
59 \( 1 + (537. + 537. i)T + 2.05e5iT^{2} \)
61 \( 1 + (352. + 352. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-255. - 255. i)T + 3.00e5iT^{2} \)
71 \( 1 + 59.2T + 3.57e5T^{2} \)
73 \( 1 + 370.T + 3.89e5T^{2} \)
79 \( 1 + 554. iT - 4.93e5T^{2} \)
83 \( 1 + (53.1 - 53.1i)T - 5.71e5iT^{2} \)
89 \( 1 + 606.T + 7.04e5T^{2} \)
97 \( 1 - 588. iT - 9.12e5T^{2} \)
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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

−11.12447675329020699496662217997, −10.25423601738409543754755393180, −9.768457234852611640695738863537, −8.074227898625226366237186546595, −7.76253358626155709047410017576, −6.52587471368489260661816400873, −5.22058213845153631254453987348, −4.59719125960444710592499212375, −3.31772539856780089150863721547, −1.73087728800687707713143852501, 0.02646095592652179694789414598, 1.57809354847863173333498257757, 2.95875392476094412709542470866, 4.44126161078406595965185121009, 5.50297578608027302268991099164, 6.31842459826671300098846894171, 7.47725156699738616399030327207, 8.262427980276211250516874639472, 9.375025363473982560182017414128, 10.03452187018260625353855383461

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