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Properties

Label 2-448-7.2-c3-0-36
Degree $2$
Conductor $448$
Sign $-0.857 + 0.514i$
Analytic cond. $26.4328$
Root an. cond. $5.14128$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (3.5 − 6.06i)5-s + (−10 + 15.5i)7-s + (13 − 22.5i)9-s + (17.5 + 30.3i)11-s − 66·13-s − 7·15-s + (−29.5 − 51.0i)17-s + (68.5 − 118. i)19-s + (18.5 + 0.866i)21-s + (3.5 − 6.06i)23-s + (38 + 65.8i)25-s − 53·27-s − 106·29-s + (−37.5 − 64.9i)31-s + ⋯
L(s)  = 1  + (−0.0962 − 0.166i)3-s + (0.313 − 0.542i)5-s + (−0.539 + 0.841i)7-s + (0.481 − 0.833i)9-s + (0.479 + 0.830i)11-s − 1.40·13-s − 0.120·15-s + (−0.420 − 0.728i)17-s + (0.827 − 1.43i)19-s + (0.192 + 0.00899i)21-s + (0.0317 − 0.0549i)23-s + (0.303 + 0.526i)25-s − 0.377·27-s − 0.678·29-s + (−0.217 − 0.376i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.514i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.857 + 0.514i$
Analytic conductor: \(26.4328\)
Root analytic conductor: \(5.14128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :3/2),\ -0.857 + 0.514i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7594649549\)
\(L(\frac12)\) \(\approx\) \(0.7594649549\)
\(L(\frac{5}{2})\) 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 \( 1 + (10 - 15.5i)T \)
good3 \( 1 + (0.5 + 0.866i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-3.5 + 6.06i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-17.5 - 30.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 66T + 2.19e3T^{2} \)
17 \( 1 + (29.5 + 51.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-68.5 + 118. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-3.5 + 6.06i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 106T + 2.43e4T^{2} \)
31 \( 1 + (37.5 + 64.9i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-5.5 + 9.52i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 498T + 6.89e4T^{2} \)
43 \( 1 + 260T + 7.95e4T^{2} \)
47 \( 1 + (-85.5 + 148. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (208.5 + 361. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (8.5 + 14.7i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-25.5 + 44.1i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-219.5 - 380. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 784T + 3.57e5T^{2} \)
73 \( 1 + (147.5 + 255. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-247.5 + 428. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 932T + 5.71e5T^{2} \)
89 \( 1 + (-436.5 + 756. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 290T + 9.12e5T^{2} \)
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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

−9.864201514040859673214246233555, −9.493652302203961949024158815543, −8.819525401778178729083425959504, −7.19606217258115330054412424780, −6.79608679876216325789891257648, −5.39603767843442473432877738175, −4.66842450250741349472286941427, −3.11407821103388558701433272857, −1.85845229305303469717841355895, −0.23658974845497600656600075901, 1.62147356057667283194297809871, 3.10674875510479317280311650510, 4.16547857462374413790499642445, 5.37226073145584825611747575685, 6.48441530671718165315003272720, 7.30447911539985007979413167220, 8.193778732281029893274575391198, 9.567745360827532662296450225576, 10.22902683996460984125710439884, 10.76654191924019140276007456802

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