↑

Properties

Label 2-224-56.51-c2-0-6
Degree $2$
Conductor $224$
Sign $0.870 - 0.492i$
Analytic cond. $6.10355$
Root an. cond. $2.47053$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.824 + 1.42i)3-s + (3.95 + 2.28i)5-s + (6.75 − 1.83i)7-s + (3.14 − 5.43i)9-s + (−6.18 − 10.7i)11-s + 18.3i·13-s + 7.52i·15-s + (6.51 + 11.2i)17-s + (−1.51 + 2.61i)19-s + (8.18 + 8.13i)21-s + (26.2 + 15.1i)23-s + (−2.09 − 3.63i)25-s + 25.1·27-s − 22.7i·29-s + (−19.5 + 11.2i)31-s + ⋯
L(s)  = 1  + (0.274 + 0.475i)3-s + (0.790 + 0.456i)5-s + (0.965 − 0.262i)7-s + (0.348 − 0.604i)9-s + (−0.562 − 0.974i)11-s + 1.41i·13-s + 0.501i·15-s + (0.383 + 0.663i)17-s + (−0.0796 + 0.137i)19-s + (0.389 + 0.387i)21-s + (1.14 + 0.659i)23-s + (−0.0839 − 0.145i)25-s + 0.933·27-s − 0.785i·29-s + (−0.630 + 0.363i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.870 - 0.492i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ 0.870 - 0.492i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.03072 + 0.535137i\)
\(L(\frac12)\) \(\approx\) \(2.03072 + 0.535137i\)
\(L(2)\) 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 \( 1 + (-6.75 + 1.83i)T \)
good3 \( 1 + (-0.824 - 1.42i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-3.95 - 2.28i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (6.18 + 10.7i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 18.3iT - 169T^{2} \)
17 \( 1 + (-6.51 - 11.2i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (1.51 - 2.61i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-26.2 - 15.1i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 22.7iT - 841T^{2} \)
31 \( 1 + (19.5 - 11.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-11.9 - 6.88i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 60.5T + 1.68e3T^{2} \)
43 \( 1 + 39.0T + 1.84e3T^{2} \)
47 \( 1 + (-17.6 - 10.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-4.12 + 2.38i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-5.86 - 10.1i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (94.3 + 54.4i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (39.5 + 68.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 12.9iT - 5.04e3T^{2} \)
73 \( 1 + (-49.2 - 85.3i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (113. + 65.6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 28.3T + 6.88e3T^{2} \)
89 \( 1 + (-78.7 + 136. i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 39.6T + 9.40e3T^{2} \)
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

−11.92479853463834360626407560872, −11.01257182076370299702796680089, −10.18465819735422451136648722014, −9.234446316767541394567944393988, −8.302670054254598760654376659293, −6.98964442851767650136113318324, −5.91246979722341044817880388599, −4.62019789904522893638515368839, −3.34922010994147432579079615024, −1.67326049554080015315210636667, 1.46634765119382521410845708182, 2.65867871163712756028765098453, 4.89602454468035097586213032394, 5.39199311654564423324809616025, 7.13843317390929738254608326084, 7.908611463690529153015270559521, 8.889633825267240025676891962367, 10.08059193636000756091900093252, 10.83506098626902858408133775141, 12.20361361157863225576958343995

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