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Properties

Label 2-448-112.13-c2-0-18
Degree $2$
Conductor $448$
Sign $0.747 - 0.664i$
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  + (−0.752 + 0.752i)3-s + (6.01 + 6.01i)5-s + (4.46 − 5.39i)7-s + 7.86i·9-s + (6.94 − 6.94i)11-s + (12.7 − 12.7i)13-s − 9.05·15-s − 5.91i·17-s + (−5.24 + 5.24i)19-s + (0.696 + 7.41i)21-s + 25.3i·23-s + 47.4i·25-s + (−12.6 − 12.6i)27-s + (−5.79 − 5.79i)29-s − 26.4i·31-s + ⋯
L(s)  = 1  + (−0.250 + 0.250i)3-s + (1.20 + 1.20i)5-s + (0.637 − 0.770i)7-s + 0.874i·9-s + (0.631 − 0.631i)11-s + (0.981 − 0.981i)13-s − 0.603·15-s − 0.348i·17-s + (−0.275 + 0.275i)19-s + (0.0331 + 0.352i)21-s + 1.10i·23-s + 1.89i·25-s + (−0.469 − 0.469i)27-s + (−0.199 − 0.199i)29-s − 0.852i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.747 - 0.664i$
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: \((2,\ 448,\ (\ :1),\ 0.747 - 0.664i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.242450834\)
\(L(\frac12)\) \(\approx\) \(2.242450834\)
\(L(2)\) 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 \( 1 + (-4.46 + 5.39i)T \)
good3 \( 1 + (0.752 - 0.752i)T - 9iT^{2} \)
5 \( 1 + (-6.01 - 6.01i)T + 25iT^{2} \)
11 \( 1 + (-6.94 + 6.94i)T - 121iT^{2} \)
13 \( 1 + (-12.7 + 12.7i)T - 169iT^{2} \)
17 \( 1 + 5.91iT - 289T^{2} \)
19 \( 1 + (5.24 - 5.24i)T - 361iT^{2} \)
23 \( 1 - 25.3iT - 529T^{2} \)
29 \( 1 + (5.79 + 5.79i)T + 841iT^{2} \)
31 \( 1 + 26.4iT - 961T^{2} \)
37 \( 1 + (3.51 - 3.51i)T - 1.36e3iT^{2} \)
41 \( 1 - 59.2T + 1.68e3T^{2} \)
43 \( 1 + (1.69 - 1.69i)T - 1.84e3iT^{2} \)
47 \( 1 - 81.1iT - 2.20e3T^{2} \)
53 \( 1 + (13.0 - 13.0i)T - 2.80e3iT^{2} \)
59 \( 1 + (-20.7 - 20.7i)T + 3.48e3iT^{2} \)
61 \( 1 + (21.5 - 21.5i)T - 3.72e3iT^{2} \)
67 \( 1 + (40.2 + 40.2i)T + 4.48e3iT^{2} \)
71 \( 1 - 99.6iT - 5.04e3T^{2} \)
73 \( 1 - 130.T + 5.32e3T^{2} \)
79 \( 1 + 116.T + 6.24e3T^{2} \)
83 \( 1 + (-31.6 + 31.6i)T - 6.88e3iT^{2} \)
89 \( 1 + 133.T + 7.92e3T^{2} \)
97 \( 1 + 27.7iT - 9.40e3T^{2} \)
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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

−11.10541928408326037248885861051, −10.26207663343078794878121959224, −9.469322809399903734910459453842, −8.119466032615087288027918521015, −7.29484037878390761029416784605, −6.08882027064857814872702506891, −5.55761157776101559499080000737, −4.07606486693643219027556725600, −2.83392286259246981920549938518, −1.43305813151199795413122564880, 1.20643520905663339437367407627, 2.06729286217262881384794100116, 4.11343250003583565366026567283, 5.11483436994875360582958948878, 6.08562710549400593869097040532, 6.72596318051523031268392549550, 8.457533791036781316624027547888, 8.991248761452723398899860113212, 9.565928199735806440334334055344, 10.84445839660715761965444445189

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