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Properties

Label 2-448-112.13-c2-0-17
Degree $2$
Conductor $448$
Sign $0.989 - 0.144i$
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  + (2.42 − 2.42i)3-s + (4.90 + 4.90i)5-s + (−5.19 + 4.68i)7-s − 2.74i·9-s + (11.4 − 11.4i)11-s + (−0.233 + 0.233i)13-s + 23.7·15-s + 13.2i·17-s + (16.1 − 16.1i)19-s + (−1.23 + 23.9i)21-s + 37.7i·23-s + 23.1i·25-s + (15.1 + 15.1i)27-s + (23.3 + 23.3i)29-s − 38.2i·31-s + ⋯
L(s)  = 1  + (0.807 − 0.807i)3-s + (0.981 + 0.981i)5-s + (−0.742 + 0.669i)7-s − 0.304i·9-s + (1.03 − 1.03i)11-s + (−0.0179 + 0.0179i)13-s + 1.58·15-s + 0.780i·17-s + (0.851 − 0.851i)19-s + (−0.0588 + 1.14i)21-s + 1.64i·23-s + 0.927i·25-s + (0.561 + 0.561i)27-s + (0.805 + 0.805i)29-s − 1.23i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.989 - 0.144i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ 0.989 - 0.144i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.647818511\)
\(L(\frac12)\) \(\approx\) \(2.647818511\)
\(L(2)\) 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 \( 1 + (5.19 - 4.68i)T \)
good3 \( 1 + (-2.42 + 2.42i)T - 9iT^{2} \)
5 \( 1 + (-4.90 - 4.90i)T + 25iT^{2} \)
11 \( 1 + (-11.4 + 11.4i)T - 121iT^{2} \)
13 \( 1 + (0.233 - 0.233i)T - 169iT^{2} \)
17 \( 1 - 13.2iT - 289T^{2} \)
19 \( 1 + (-16.1 + 16.1i)T - 361iT^{2} \)
23 \( 1 - 37.7iT - 529T^{2} \)
29 \( 1 + (-23.3 - 23.3i)T + 841iT^{2} \)
31 \( 1 + 38.2iT - 961T^{2} \)
37 \( 1 + (-5.37 + 5.37i)T - 1.36e3iT^{2} \)
41 \( 1 - 13.3T + 1.68e3T^{2} \)
43 \( 1 + (27.7 - 27.7i)T - 1.84e3iT^{2} \)
47 \( 1 + 33.0iT - 2.20e3T^{2} \)
53 \( 1 + (9.52 - 9.52i)T - 2.80e3iT^{2} \)
59 \( 1 + (46.0 + 46.0i)T + 3.48e3iT^{2} \)
61 \( 1 + (-58.0 + 58.0i)T - 3.72e3iT^{2} \)
67 \( 1 + (30.9 + 30.9i)T + 4.48e3iT^{2} \)
71 \( 1 - 4.71iT - 5.04e3T^{2} \)
73 \( 1 + 74.6T + 5.32e3T^{2} \)
79 \( 1 + 32.5T + 6.24e3T^{2} \)
83 \( 1 + (66.2 - 66.2i)T - 6.88e3iT^{2} \)
89 \( 1 - 15.6T + 7.92e3T^{2} \)
97 \( 1 + 4.30iT - 9.40e3T^{2} \)
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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

−10.95267928406287121360930600726, −9.740895961416775955814493626754, −9.176203215145848051542315382839, −8.214437777882638712154464311779, −7.05283833824565777595348893237, −6.37207719690137964506530214916, −5.56801631964153178040996909051, −3.43738695909154360779930752447, −2.73831572587192723656580532024, −1.54876339409219883849318615543, 1.19303585198563376844815501320, 2.79449621714738997952495947258, 4.07916707022286909179978907192, 4.79348771408174608542524208502, 6.16628409613561829513530099446, 7.12805016584784848329653419496, 8.537164080515570841150431479257, 9.227518645074617609336040602680, 9.880491001803668680599102062971, 10.26878672638783615196275707957

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