↑

Properties

Label 2-684-228.227-c2-0-21
Degree $2$
Conductor $684$
Sign $0.100 - 0.994i$
Analytic cond. $18.6376$
Root an. cond. $4.31713$
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  + (−1.45 − 1.37i)2-s + (0.211 + 3.99i)4-s + 4.00i·5-s + 12.0i·7-s + (5.19 − 6.08i)8-s + (5.50 − 5.80i)10-s + 15.7·11-s + 4.74i·13-s + (16.5 − 17.4i)14-s + (−15.9 + 1.68i)16-s + 7.81i·17-s + (13.2 − 13.6i)19-s + (−15.9 + 0.846i)20-s + (−22.8 − 21.6i)22-s + 3.71·23-s + ⋯
L(s)  = 1  + (−0.725 − 0.688i)2-s + (0.0528 + 0.998i)4-s + 0.800i·5-s + 1.71i·7-s + (0.648 − 0.760i)8-s + (0.550 − 0.580i)10-s + 1.43·11-s + 0.364i·13-s + (1.18 − 1.24i)14-s + (−0.994 + 0.105i)16-s + 0.459i·17-s + (0.695 − 0.718i)19-s + (−0.799 + 0.0423i)20-s + (−1.03 − 0.985i)22-s + 0.161·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.100 - 0.994i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ 0.100 - 0.994i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.257563295\)
\(L(\frac12)\) \(\approx\) \(1.257563295\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.45 + 1.37i)T \)
3 \( 1 \)
19 \( 1 + (-13.2 + 13.6i)T \)
good5 \( 1 - 4.00iT - 25T^{2} \)
7 \( 1 - 12.0iT - 49T^{2} \)
11 \( 1 - 15.7T + 121T^{2} \)
13 \( 1 - 4.74iT - 169T^{2} \)
17 \( 1 - 7.81iT - 289T^{2} \)
23 \( 1 - 3.71T + 529T^{2} \)
29 \( 1 - 46.8T + 841T^{2} \)
31 \( 1 + 56.2T + 961T^{2} \)
37 \( 1 - 52.6iT - 1.36e3T^{2} \)
41 \( 1 + 31.8T + 1.68e3T^{2} \)
43 \( 1 - 30.9iT - 1.84e3T^{2} \)
47 \( 1 + 35.5T + 2.20e3T^{2} \)
53 \( 1 - 64.9T + 2.80e3T^{2} \)
59 \( 1 + 31.1iT - 3.48e3T^{2} \)
61 \( 1 - 6.13T + 3.72e3T^{2} \)
67 \( 1 + 102.T + 4.48e3T^{2} \)
71 \( 1 + 89.2iT - 5.04e3T^{2} \)
73 \( 1 - 101.T + 5.32e3T^{2} \)
79 \( 1 - 26.6T + 6.24e3T^{2} \)
83 \( 1 + 62.4T + 6.88e3T^{2} \)
89 \( 1 + 17.0T + 7.92e3T^{2} \)
97 \( 1 - 134. iT - 9.40e3T^{2} \)
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

−10.48666158349949522052951780177, −9.434045949829209522063442093878, −9.000208978674663832431439909197, −8.201966426923930355716635394083, −6.89625866717579061603341067795, −6.34844164925286614645090626854, −4.90745373270418691854536925189, −3.48760483849766628864052445449, −2.66562910019132897373734682699, −1.52236594408910990158872831855, 0.65507171238842385686287668865, 1.42784953397223493974171083141, 3.70882757781444606893822299648, 4.63737737233311065818269430872, 5.67359350527915067715043798027, 6.90133862580209222192008745632, 7.30750724684566008021348135102, 8.383205220641798875267695934547, 9.135481631527526083741221687432, 9.948964252720486467046966230158

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