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Properties

Label 2-448-112.27-c1-0-4
Degree $2$
Conductor $448$
Sign $0.621 - 0.783i$
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  + (1.03 − 1.03i)3-s + (−1.68 + 1.68i)5-s + (−1.38 + 2.25i)7-s + 0.851i·9-s + (2 − 2i)11-s + (4.80 + 4.80i)13-s + 3.49i·15-s + 1.13i·17-s + (−1.21 + 1.21i)19-s + (0.900 + 3.77i)21-s + 1.33·23-s − 0.690i·25-s + (3.99 + 3.99i)27-s + (5.26 − 5.26i)29-s − 8.31·31-s + ⋯
L(s)  = 1  + (0.598 − 0.598i)3-s + (−0.754 + 0.754i)5-s + (−0.523 + 0.851i)7-s + 0.283i·9-s + (0.603 − 0.603i)11-s + (1.33 + 1.33i)13-s + 0.902i·15-s + 0.275i·17-s + (−0.279 + 0.279i)19-s + (0.196 + 0.823i)21-s + 0.278·23-s − 0.138i·25-s + (0.768 + 0.768i)27-s + (0.978 − 0.978i)29-s − 1.49·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.621 - 0.783i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.621 - 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28358 + 0.620353i\)
\(L(\frac12)\) \(\approx\) \(1.28358 + 0.620353i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.38 - 2.25i)T \)
good3 \( 1 + (-1.03 + 1.03i)T - 3iT^{2} \)
5 \( 1 + (1.68 - 1.68i)T - 5iT^{2} \)
11 \( 1 + (-2 + 2i)T - 11iT^{2} \)
13 \( 1 + (-4.80 - 4.80i)T + 13iT^{2} \)
17 \( 1 - 1.13iT - 17T^{2} \)
19 \( 1 + (1.21 - 1.21i)T - 19iT^{2} \)
23 \( 1 - 1.33T + 23T^{2} \)
29 \( 1 + (-5.26 + 5.26i)T - 29iT^{2} \)
31 \( 1 + 8.31T + 31T^{2} \)
37 \( 1 + (4.18 + 4.18i)T + 37iT^{2} \)
41 \( 1 + 1.63T + 41T^{2} \)
43 \( 1 + (-1.33 + 1.33i)T - 43iT^{2} \)
47 \( 1 - 1.93T + 47T^{2} \)
53 \( 1 + (-6.34 - 6.34i)T + 53iT^{2} \)
59 \( 1 + (-3.29 - 3.29i)T + 59iT^{2} \)
61 \( 1 + (2.04 + 2.04i)T + 61iT^{2} \)
67 \( 1 + (0.107 + 0.107i)T + 67iT^{2} \)
71 \( 1 + 13.0T + 71T^{2} \)
73 \( 1 + 6.24T + 73T^{2} \)
79 \( 1 - 4.51iT - 79T^{2} \)
83 \( 1 + (-9.71 + 9.71i)T - 83iT^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 + 3.23iT - 97T^{2} \)
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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

−11.33777565671249955442103066850, −10.50767171404151511668469692415, −8.958708927616188585344766170359, −8.699116507815565329005852836754, −7.52154101042544963310483819942, −6.67471865241298012088768708879, −5.85400908111333711399134985864, −4.04997492804769286175764408972, −3.16193430909162295872297912011, −1.86550250507541212340547307237, 0.907601649639233487005939117788, 3.29701810697850506536171614244, 3.88618209646890317006153145244, 4.90327246632577324681272104517, 6.37605401574456047608773119772, 7.40046113969284079425273858552, 8.506229119880000694733019513092, 8.992062971414613485393918059511, 10.08439306989848501730808411592, 10.75855498788760670720694204324

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