↑

Properties

Label 2-448-7.4-c3-0-37
Degree $2$
Conductor $448$
Sign $-0.386 + 0.922i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.5 − 4.33i)3-s + (−4.5 − 7.79i)5-s + (14 − 12.1i)7-s + (0.999 + 1.73i)9-s + (28.5 − 49.3i)11-s + 70·13-s − 45.0·15-s + (−25.5 + 44.1i)17-s + (−2.5 − 4.33i)19-s + (−17.5 − 90.9i)21-s + (34.5 + 59.7i)23-s + (22 − 38.1i)25-s + 144.·27-s − 114·29-s + (11.5 − 19.9i)31-s + ⋯
L(s)  = 1  + (0.481 − 0.833i)3-s + (−0.402 − 0.697i)5-s + (0.755 − 0.654i)7-s + (0.0370 + 0.0641i)9-s + (0.781 − 1.35i)11-s + 1.49·13-s − 0.774·15-s + (−0.363 + 0.630i)17-s + (−0.0301 − 0.0522i)19-s + (−0.181 − 0.944i)21-s + (0.312 + 0.541i)23-s + (0.175 − 0.304i)25-s + 1.03·27-s − 0.729·29-s + (0.0666 − 0.115i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(26.4328\)
Root analytic conductor: \(5.14128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :3/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.560464619\)
\(L(\frac12)\) \(\approx\) \(2.560464619\)
\(L(\frac{5}{2})\) 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 \( 1 + (-14 + 12.1i)T \)
good3 \( 1 + (-2.5 + 4.33i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (4.5 + 7.79i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-28.5 + 49.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 70T + 2.19e3T^{2} \)
17 \( 1 + (25.5 - 44.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-34.5 - 59.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 114T + 2.43e4T^{2} \)
31 \( 1 + (-11.5 + 19.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (126.5 + 219. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 42T + 6.89e4T^{2} \)
43 \( 1 + 124T + 7.95e4T^{2} \)
47 \( 1 + (-100.5 - 174. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (196.5 - 340. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (109.5 - 189. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (354.5 + 614. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (209.5 - 362. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 96T + 3.57e5T^{2} \)
73 \( 1 + (-156.5 + 271. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-230.5 - 399. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 588T + 5.71e5T^{2} \)
89 \( 1 + (-508.5 - 880. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.83e3T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.79507434993760148570222767979, −9.066101887355944322661290135830, −8.433879334582861009009730320132, −7.888621322980113470801209438367, −6.78618960292018379074763073053, −5.75888343680534002836143195290, −4.36648112642671428464989556482, −3.46819808282200579434804824256, −1.63519928139954108565444602128, −0.881028633572414486706396969942, 1.63357758031644338796758413874, 3.10070034197688912828502902709, 4.06766823395712928197477679939, 4.94596640679020762080931486704, 6.41294352376036929593122995993, 7.25547657039492593227310214135, 8.511682749539919295050128580981, 9.100884932056574081471667118167, 10.01936480417154492860577475397, 10.97430071428683296978443838944

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