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Properties

Label 2-448-56.51-c2-0-5
Degree $2$
Conductor $448$
Sign $0.562 - 0.826i$
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  + (−0.565 − 0.980i)3-s + (3.20 + 1.84i)5-s + (−2.72 + 6.44i)7-s + (3.85 − 6.68i)9-s + (3.28 + 5.69i)11-s + 1.88i·13-s − 4.18i·15-s + (−0.399 − 0.692i)17-s + (−14.6 + 25.3i)19-s + (7.86 − 0.982i)21-s + (22.4 + 12.9i)23-s + (−5.66 − 9.81i)25-s − 18.9·27-s + 38.2i·29-s + (6.26 − 3.61i)31-s + ⋯
L(s)  = 1  + (−0.188 − 0.326i)3-s + (0.640 + 0.369i)5-s + (−0.388 + 0.921i)7-s + (0.428 − 0.742i)9-s + (0.298 + 0.517i)11-s + 0.145i·13-s − 0.278i·15-s + (−0.0235 − 0.0407i)17-s + (−0.769 + 1.33i)19-s + (0.374 − 0.0467i)21-s + (0.974 + 0.562i)23-s + (−0.226 − 0.392i)25-s − 0.700·27-s + 1.31i·29-s + (0.201 − 0.116i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 - 0.826i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.562 - 0.826i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ 0.562 - 0.826i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.668628327\)
\(L(\frac12)\) \(\approx\) \(1.668628327\)
\(L(2)\) 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 \( 1 + (2.72 - 6.44i)T \)
good3 \( 1 + (0.565 + 0.980i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-3.20 - 1.84i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-3.28 - 5.69i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 1.88iT - 169T^{2} \)
17 \( 1 + (0.399 + 0.692i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (14.6 - 25.3i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-22.4 - 12.9i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 38.2iT - 841T^{2} \)
31 \( 1 + (-6.26 + 3.61i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-56.7 - 32.7i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 55.0T + 1.68e3T^{2} \)
43 \( 1 + 22.3T + 1.84e3T^{2} \)
47 \( 1 + (-16.3 - 9.42i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-0.264 + 0.152i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-27.0 - 46.9i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-28.4 - 16.4i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-4.74 - 8.21i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 33.4iT - 5.04e3T^{2} \)
73 \( 1 + (-47.2 - 81.9i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-7.35 - 4.24i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 77.8T + 6.88e3T^{2} \)
89 \( 1 + (-23.8 + 41.3i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 125.T + 9.40e3T^{2} \)
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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

−11.06267671560511599839265916597, −9.901487311847262598163990699380, −9.439755294904244015383002078637, −8.381981227702168582528646705934, −7.08298327303354419025415537667, −6.33459255096912137463032874214, −5.60791745919772801982295959042, −4.13034998470181819405447146013, −2.76925451693201186722624293729, −1.49468160963279661791176461072, 0.76414134812781462506902275706, 2.44324869813297319235419082429, 4.02246945280920363684109788598, 4.86558175098277326971786992352, 6.01049046371396807563477984384, 6.97705628949237324622831256875, 7.995222009324494448683785951266, 9.172800326839336489621094161486, 9.826279473522112912809580356606, 10.82273631081431165975113163283

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