↑

Properties

Label 2-224-32.11-c2-0-24
Degree $2$
Conductor $224$
Sign $0.967 + 0.252i$
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  + (−1.99 − 0.180i)2-s + (0.825 + 1.99i)3-s + (3.93 + 0.718i)4-s + (0.360 − 0.870i)5-s + (−1.28 − 4.11i)6-s + (1.87 − 1.87i)7-s + (−7.70 − 2.14i)8-s + (3.07 − 3.07i)9-s + (−0.875 + 1.66i)10-s + (5.69 − 13.7i)11-s + (1.81 + 8.43i)12-s + (2.01 + 4.86i)13-s + (−4.06 + 3.38i)14-s + 2.03·15-s + (14.9 + 5.65i)16-s − 19.7i·17-s + ⋯
L(s)  = 1  + (−0.995 − 0.0901i)2-s + (0.275 + 0.664i)3-s + (0.983 + 0.179i)4-s + (0.0721 − 0.174i)5-s + (−0.214 − 0.686i)6-s + (0.267 − 0.267i)7-s + (−0.963 − 0.267i)8-s + (0.341 − 0.341i)9-s + (−0.0875 + 0.166i)10-s + (0.518 − 1.25i)11-s + (0.151 + 0.702i)12-s + (0.155 + 0.374i)13-s + (−0.290 + 0.242i)14-s + 0.135·15-s + (0.935 + 0.353i)16-s − 1.16i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.967 + 0.252i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ 0.967 + 0.252i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.19692 - 0.153491i\)
\(L(\frac12)\) \(\approx\) \(1.19692 - 0.153491i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.99 + 0.180i)T \)
7 \( 1 + (-1.87 + 1.87i)T \)
good3 \( 1 + (-0.825 - 1.99i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (-0.360 + 0.870i)T + (-17.6 - 17.6i)T^{2} \)
11 \( 1 + (-5.69 + 13.7i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (-2.01 - 4.86i)T + (-119. + 119. i)T^{2} \)
17 \( 1 + 19.7iT - 289T^{2} \)
19 \( 1 + (23.8 - 9.86i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (-18.8 - 18.8i)T + 529iT^{2} \)
29 \( 1 + (-45.5 + 18.8i)T + (594. - 594. i)T^{2} \)
31 \( 1 + 37.3iT - 961T^{2} \)
37 \( 1 + (-19.3 + 46.8i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + (16.8 - 16.8i)T - 1.68e3iT^{2} \)
43 \( 1 + (31.7 - 76.5i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 83.8T + 2.20e3T^{2} \)
53 \( 1 + (25.3 + 10.4i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (37.1 + 15.3i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (47.4 - 19.6i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-31.7 - 76.5i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (22.6 - 22.6i)T - 5.04e3iT^{2} \)
73 \( 1 + (-37.4 + 37.4i)T - 5.32e3iT^{2} \)
79 \( 1 + 11.5T + 6.24e3T^{2} \)
83 \( 1 + (147. - 61.2i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (103. + 103. i)T + 7.92e3iT^{2} \)
97 \( 1 + 82.5T + 9.40e3T^{2} \)
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

−11.54326071178593831116118687146, −10.96262078354046117387306971526, −9.836896166771002390088799074345, −9.122195278644615361036659574225, −8.353194758937714798924774518459, −7.09620443436178238278248562309, −6.01948389762228515973138823749, −4.29184148102393602566401301329, −3.00327146056871219342134600611, −1.03755935625241107300597286972, 1.45198385462881265621660583139, 2.58477416981243750884889680098, 4.70901129242477933340959573638, 6.52916506817892527011703389841, 6.99388049535179752493924518659, 8.293269542092676479902417235463, 8.794233409758203641831383150991, 10.34413106946607476361131406054, 10.64444575622410871034485901232, 12.33274252443817151617279631096

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