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Properties

Label 2-448-112.19-c1-0-2
Degree $2$
Conductor $448$
Sign $-0.923 - 0.382i$
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  + (−0.615 + 2.29i)3-s + (−0.835 + 0.223i)5-s + (1.25 + 2.32i)7-s + (−2.29 − 1.32i)9-s + (−0.544 + 2.03i)11-s + (0.336 − 0.336i)13-s − 2.05i·15-s + (−0.0488 + 0.0282i)17-s + (−7.89 + 2.11i)19-s + (−6.11 + 1.45i)21-s + (3.59 − 6.23i)23-s + (−3.68 + 2.12i)25-s + (−0.584 + 0.584i)27-s + (4.81 + 4.81i)29-s + (−1.84 − 3.20i)31-s + ⋯
L(s)  = 1  + (−0.355 + 1.32i)3-s + (−0.373 + 0.100i)5-s + (0.475 + 0.879i)7-s + (−0.765 − 0.442i)9-s + (−0.164 + 0.613i)11-s + (0.0933 − 0.0933i)13-s − 0.531i·15-s + (−0.0118 + 0.00684i)17-s + (−1.81 + 0.485i)19-s + (−1.33 + 0.317i)21-s + (0.750 − 1.29i)23-s + (−0.736 + 0.425i)25-s + (−0.112 + 0.112i)27-s + (0.894 + 0.894i)29-s + (−0.332 − 0.575i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.923 - 0.382i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ -0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.185469 + 0.932731i\)
\(L(\frac12)\) \(\approx\) \(0.185469 + 0.932731i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.25 - 2.32i)T \)
good3 \( 1 + (0.615 - 2.29i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (0.835 - 0.223i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.544 - 2.03i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.336 + 0.336i)T - 13iT^{2} \)
17 \( 1 + (0.0488 - 0.0282i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.89 - 2.11i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.59 + 6.23i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.81 - 4.81i)T + 29iT^{2} \)
31 \( 1 + (1.84 + 3.20i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.876 + 3.27i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 4.95T + 41T^{2} \)
43 \( 1 + (4.99 + 4.99i)T + 43iT^{2} \)
47 \( 1 + (5.50 - 9.53i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.62 - 0.971i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-8.49 - 2.27i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.66 - 6.21i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-10.4 - 2.80i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 13.1T + 71T^{2} \)
73 \( 1 + (-2.69 - 4.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.99 - 2.88i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.52 - 4.52i)T + 83iT^{2} \)
89 \( 1 + (1.28 - 2.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 9.06iT - 97T^{2} \)
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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

−11.17216300512880960875642474310, −10.69849964667772664387207103690, −9.798808049507530024032918761390, −8.862252321676753468654699018275, −8.123224082922089169882933998911, −6.71968274413637863873566953582, −5.55814231553481165953502634669, −4.69390537125136830419691720212, −3.88689527202520065153385114373, −2.34185189321719165932595251514, 0.61884854047589373516563149172, 2.00494863757900113953630831501, 3.73644578816469643680762261543, 4.97244046637175922390980252735, 6.30930645226460045265643159956, 6.93871540281750727449313196209, 7.937269131729260062539638235004, 8.457046051547106021994169023631, 9.939620794216146639852265625460, 11.06826907281966733764558564105

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