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Properties

Label 2-448-112.27-c3-0-12
Degree $2$
Conductor $448$
Sign $0.0269 - 0.999i$
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  + (4.79 − 4.79i)3-s + (2.04 − 2.04i)5-s + (−13.3 + 12.8i)7-s − 19.0i·9-s + (−34.5 + 34.5i)11-s + (−7.32 − 7.32i)13-s − 19.6i·15-s + 133. i·17-s + (13.6 − 13.6i)19-s + (−2.69 + 125. i)21-s − 129.·23-s + 116. i·25-s + (38.2 + 38.2i)27-s + (85.7 − 85.7i)29-s − 239.·31-s + ⋯
L(s)  = 1  + (0.923 − 0.923i)3-s + (0.183 − 0.183i)5-s + (−0.722 + 0.691i)7-s − 0.704i·9-s + (−0.947 + 0.947i)11-s + (−0.156 − 0.156i)13-s − 0.337i·15-s + 1.90i·17-s + (0.165 − 0.165i)19-s + (−0.0280 + 1.30i)21-s − 1.17·23-s + 0.933i·25-s + (0.272 + 0.272i)27-s + (0.549 − 0.549i)29-s − 1.38·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0269 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0269 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.0269 - 0.999i$
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: \((2,\ 448,\ (\ :3/2),\ 0.0269 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.312640044\)
\(L(\frac12)\) \(\approx\) \(1.312640044\)
\(L(\frac{5}{2})\) 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 \( 1 + (13.3 - 12.8i)T \)
good3 \( 1 + (-4.79 + 4.79i)T - 27iT^{2} \)
5 \( 1 + (-2.04 + 2.04i)T - 125iT^{2} \)
11 \( 1 + (34.5 - 34.5i)T - 1.33e3iT^{2} \)
13 \( 1 + (7.32 + 7.32i)T + 2.19e3iT^{2} \)
17 \( 1 - 133. iT - 4.91e3T^{2} \)
19 \( 1 + (-13.6 + 13.6i)T - 6.85e3iT^{2} \)
23 \( 1 + 129.T + 1.21e4T^{2} \)
29 \( 1 + (-85.7 + 85.7i)T - 2.43e4iT^{2} \)
31 \( 1 + 239.T + 2.97e4T^{2} \)
37 \( 1 + (50.7 + 50.7i)T + 5.06e4iT^{2} \)
41 \( 1 - 118.T + 6.89e4T^{2} \)
43 \( 1 + (-75.4 + 75.4i)T - 7.95e4iT^{2} \)
47 \( 1 + 169.T + 1.03e5T^{2} \)
53 \( 1 + (-482. - 482. i)T + 1.48e5iT^{2} \)
59 \( 1 + (285. + 285. i)T + 2.05e5iT^{2} \)
61 \( 1 + (313. + 313. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-58.2 - 58.2i)T + 3.00e5iT^{2} \)
71 \( 1 - 374.T + 3.57e5T^{2} \)
73 \( 1 - 610.T + 3.89e5T^{2} \)
79 \( 1 + 460. iT - 4.93e5T^{2} \)
83 \( 1 + (875. - 875. i)T - 5.71e5iT^{2} \)
89 \( 1 + 633.T + 7.04e5T^{2} \)
97 \( 1 - 946. iT - 9.12e5T^{2} \)
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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

−10.75693299054901095285444095094, −9.855197971114826224853166091399, −8.972950148497842853764088850987, −8.092307078938678598665513165559, −7.43096483506904223638606777507, −6.31845339661501271025994028451, −5.35275721410005966676253186220, −3.76879594750194334726074498541, −2.50364474796370093104980852190, −1.75352328209435230274401789300, 0.34387728702699672621786074023, 2.62294181190414352673073012814, 3.33903203546686801120450918206, 4.39913309126112751627191001369, 5.57982172293140895359950382624, 6.84583198064851943439785072054, 7.84393114925619760861459518231, 8.805027103640597731720723790986, 9.685127180234239825726943108466, 10.19362585649302882391907748313

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