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Properties

Label 2-448-7.6-c4-0-6
Degree $2$
Conductor $448$
Sign $0.131 + 0.991i$
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  + 12.6i·3-s + 23.1i·5-s + (6.45 + 48.5i)7-s − 79.9·9-s − 191.·11-s + 48.5i·13-s − 294.·15-s − 181. i·17-s − 599. i·19-s + (−616. + 81.9i)21-s − 469.·23-s + 86.8·25-s + 13.0i·27-s + 338.·29-s + 267. i·31-s + ⋯
L(s)  = 1  + 1.40i·3-s + 0.927i·5-s + (0.131 + 0.991i)7-s − 0.987·9-s − 1.58·11-s + 0.287i·13-s − 1.30·15-s − 0.629i·17-s − 1.66i·19-s + (−1.39 + 0.185i)21-s − 0.887·23-s + 0.138·25-s + 0.0179i·27-s + 0.402·29-s + 0.278i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 + 0.991i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.131 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.131 + 0.991i$
Analytic conductor: \(46.3097\)
Root analytic conductor: \(6.80512\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :2),\ 0.131 + 0.991i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.4892401324\)
\(L(\frac12)\) \(\approx\) \(0.4892401324\)
\(L(3)\) 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 \( 1 + (-6.45 - 48.5i)T \)
good3 \( 1 - 12.6iT - 81T^{2} \)
5 \( 1 - 23.1iT - 625T^{2} \)
11 \( 1 + 191.T + 1.46e4T^{2} \)
13 \( 1 - 48.5iT - 2.85e4T^{2} \)
17 \( 1 + 181. iT - 8.35e4T^{2} \)
19 \( 1 + 599. iT - 1.30e5T^{2} \)
23 \( 1 + 469.T + 2.79e5T^{2} \)
29 \( 1 - 338.T + 7.07e5T^{2} \)
31 \( 1 - 267. iT - 9.23e5T^{2} \)
37 \( 1 - 668.T + 1.87e6T^{2} \)
41 \( 1 - 1.32e3iT - 2.82e6T^{2} \)
43 \( 1 + 1.94e3T + 3.41e6T^{2} \)
47 \( 1 - 2.93e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.46e3T + 7.89e6T^{2} \)
59 \( 1 - 1.73e3iT - 1.21e7T^{2} \)
61 \( 1 + 246. iT - 1.38e7T^{2} \)
67 \( 1 - 1.07e3T + 2.01e7T^{2} \)
71 \( 1 + 2.27e3T + 2.54e7T^{2} \)
73 \( 1 + 7.10e3iT - 2.83e7T^{2} \)
79 \( 1 - 7.01e3T + 3.89e7T^{2} \)
83 \( 1 - 1.44e3iT - 4.74e7T^{2} \)
89 \( 1 + 2.13e3iT - 6.27e7T^{2} \)
97 \( 1 + 5.89e3iT - 8.85e7T^{2} \)
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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

−10.99918530417194037590401512309, −10.30278013871484237413521856663, −9.529161136646600611123259716383, −8.696837037769268136339743142264, −7.59698633578780489946841116054, −6.39171168200024475366341494446, −5.21897683441617812187078184472, −4.62639335634641481380409522840, −3.08404114633171324939053081703, −2.52611819754367760592639566194, 0.13716999809232537602215420346, 1.16083330769771848997412355039, 2.17283793279220840085404143029, 3.80063155314878208937102648217, 5.11526074909850544364981277598, 6.05322883155413170971135887510, 7.18491609757673151253641958585, 8.103373872555252091450000199511, 8.249168631868654206687142540617, 9.994683180321941082830523240884

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