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Properties

Label 2-448-7.3-c2-0-22
Degree $2$
Conductor $448$
Sign $0.0725 + 0.997i$
Analytic cond. $12.2071$
Root an. cond. $3.49386$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.621 − 0.358i)3-s + (5.74 − 3.31i)5-s + (6.24 − 3.16i)7-s + (−4.24 − 7.34i)9-s + (−2.37 + 4.11i)11-s − 15.2i·13-s − 4.75·15-s + (−3.25 − 1.88i)17-s + (−3.62 + 2.09i)19-s + (−5.01 − 0.271i)21-s + (13.8 + 24.0i)23-s + (9.48 − 16.4i)25-s + 12.5i·27-s − 3.51·29-s + (−42.3 − 24.4i)31-s + ⋯
L(s)  = 1  + (−0.207 − 0.119i)3-s + (1.14 − 0.663i)5-s + (0.891 − 0.452i)7-s + (−0.471 − 0.816i)9-s + (−0.216 + 0.374i)11-s − 1.17i·13-s − 0.317·15-s + (−0.191 − 0.110i)17-s + (−0.190 + 0.110i)19-s + (−0.238 − 0.0129i)21-s + (0.602 + 1.04i)23-s + (0.379 − 0.657i)25-s + 0.464i·27-s − 0.121·29-s + (−1.36 − 0.788i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.0725 + 0.997i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ 0.0725 + 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.917692017\)
\(L(\frac12)\) \(\approx\) \(1.917692017\)
\(L(2)\) 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 \( 1 + (-6.24 + 3.16i)T \)
good3 \( 1 + (0.621 + 0.358i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (-5.74 + 3.31i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (2.37 - 4.11i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 15.2iT - 169T^{2} \)
17 \( 1 + (3.25 + 1.88i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (3.62 - 2.09i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-13.8 - 24.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 3.51T + 841T^{2} \)
31 \( 1 + (42.3 + 24.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (1.47 + 2.54i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 27.9iT - 1.68e3T^{2} \)
43 \( 1 - 10.4T + 1.84e3T^{2} \)
47 \( 1 + (-45.6 + 26.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-27.9 + 48.4i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (33.5 + 19.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-78.3 + 45.2i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (17.3 - 29.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 36.4T + 5.04e3T^{2} \)
73 \( 1 + (-45.5 - 26.3i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-16.8 - 29.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 127. iT - 6.88e3T^{2} \)
89 \( 1 + (43.5 - 25.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 101. iT - 9.40e3T^{2} \)
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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

−10.68322231136547491066503361923, −9.705564343798989456677264102594, −8.979791701551711869520533024123, −7.968640808144406819314521606870, −6.94314861908577661543133487488, −5.57529551840303479045795040972, −5.29383622452446459800041626148, −3.75392381254251300740933194488, −2.11557842776077351742022583051, −0.833574378605946227783919683364, 1.83850878970497622893261786615, 2.68403720729651061795373153848, 4.51662170629242291754704518225, 5.46829243471903149497095710201, 6.27115586218404676012776254602, 7.33995247810839533894710716608, 8.567687678095709784916361178940, 9.214195106412738852979573820034, 10.50008834089043216343229744886, 10.90896803630427425482955767799

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