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Properties

Label 2-448-112.13-c2-0-4
Degree $2$
Conductor $448$
Sign $-0.467 - 0.883i$
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  + (2.28 − 2.28i)3-s + (−2.90 − 2.90i)5-s + (−2.26 + 6.62i)7-s − 1.42i·9-s + (−9.96 + 9.96i)11-s + (−15.5 + 15.5i)13-s − 13.2·15-s − 32.2i·17-s + (−14.4 + 14.4i)19-s + (9.95 + 20.2i)21-s + 16.0i·23-s − 8.10i·25-s + (17.3 + 17.3i)27-s + (−10.0 − 10.0i)29-s + 3.11i·31-s + ⋯
L(s)  = 1  + (0.760 − 0.760i)3-s + (−0.581 − 0.581i)5-s + (−0.323 + 0.946i)7-s − 0.157i·9-s + (−0.905 + 0.905i)11-s + (−1.19 + 1.19i)13-s − 0.884·15-s − 1.89i·17-s + (−0.759 + 0.759i)19-s + (0.473 + 0.966i)21-s + 0.697i·23-s − 0.324i·25-s + (0.640 + 0.640i)27-s + (−0.347 − 0.347i)29-s + 0.100i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.467 - 0.883i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.467 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.467 - 0.883i$
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: \((2,\ 448,\ (\ :1),\ -0.467 - 0.883i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5759599625\)
\(L(\frac12)\) \(\approx\) \(0.5759599625\)
\(L(2)\) 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 \( 1 + (2.26 - 6.62i)T \)
good3 \( 1 + (-2.28 + 2.28i)T - 9iT^{2} \)
5 \( 1 + (2.90 + 2.90i)T + 25iT^{2} \)
11 \( 1 + (9.96 - 9.96i)T - 121iT^{2} \)
13 \( 1 + (15.5 - 15.5i)T - 169iT^{2} \)
17 \( 1 + 32.2iT - 289T^{2} \)
19 \( 1 + (14.4 - 14.4i)T - 361iT^{2} \)
23 \( 1 - 16.0iT - 529T^{2} \)
29 \( 1 + (10.0 + 10.0i)T + 841iT^{2} \)
31 \( 1 - 3.11iT - 961T^{2} \)
37 \( 1 + (36.8 - 36.8i)T - 1.36e3iT^{2} \)
41 \( 1 - 39.5T + 1.68e3T^{2} \)
43 \( 1 + (-19.5 + 19.5i)T - 1.84e3iT^{2} \)
47 \( 1 + 20.9iT - 2.20e3T^{2} \)
53 \( 1 + (18.6 - 18.6i)T - 2.80e3iT^{2} \)
59 \( 1 + (-21.7 - 21.7i)T + 3.48e3iT^{2} \)
61 \( 1 + (-23.5 + 23.5i)T - 3.72e3iT^{2} \)
67 \( 1 + (14.0 + 14.0i)T + 4.48e3iT^{2} \)
71 \( 1 - 2.39iT - 5.04e3T^{2} \)
73 \( 1 + 22.2T + 5.32e3T^{2} \)
79 \( 1 + 67.0T + 6.24e3T^{2} \)
83 \( 1 + (82.2 - 82.2i)T - 6.88e3iT^{2} \)
89 \( 1 + 103.T + 7.92e3T^{2} \)
97 \( 1 - 18.8iT - 9.40e3T^{2} \)
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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

−11.53964500327108126028156082047, −9.983986717613683593975054987087, −9.256472038289982251133225274029, −8.396562888968928270625079030725, −7.52089233130805833889425197725, −6.92479166369648248596737348820, −5.31745740561961412845881955723, −4.48252007984927500888987631638, −2.74573475227062471832259557592, −2.00605838417748987441550269571, 0.20058682541670183912760062895, 2.74143053172912406826258672090, 3.52533988079001138742092290618, 4.41131683292169310008595007999, 5.82502817699220782211031791854, 7.08586479791569440186546133500, 7.953865549539615809373638965862, 8.677415128598601577610959740717, 9.897197683174642161716990490923, 10.61948675288985252628630295978

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