Properties

Label 2-684-171.103-c2-0-25
Degree $2$
Conductor $684$
Sign $0.927 + 0.373i$
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  + (1.98 + 2.25i)3-s + (−0.178 − 0.309i)5-s + (−3.52 − 6.09i)7-s + (−1.13 + 8.92i)9-s + (−4.98 − 8.63i)11-s − 2.10i·13-s + (0.341 − 1.01i)15-s + (7.21 − 12.5i)17-s + (18.9 + 0.100i)19-s + (6.73 − 20.0i)21-s + 45.4·23-s + (12.4 − 21.5i)25-s + (−22.3 + 15.1i)27-s + (−3.90 − 2.25i)29-s + (19.9 + 11.5i)31-s + ⋯
L(s)  = 1  + (0.661 + 0.750i)3-s + (−0.0356 − 0.0618i)5-s + (−0.502 − 0.870i)7-s + (−0.125 + 0.992i)9-s + (−0.453 − 0.785i)11-s − 0.161i·13-s + (0.0227 − 0.0676i)15-s + (0.424 − 0.735i)17-s + (0.999 + 0.00526i)19-s + (0.320 − 0.953i)21-s + 1.97·23-s + (0.497 − 0.861i)25-s + (−0.827 + 0.561i)27-s + (−0.134 − 0.0778i)29-s + (0.643 + 0.371i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.927 + 0.373i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ 0.927 + 0.373i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.091464764\)
\(L(\frac12)\) \(\approx\) \(2.091464764\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.98 - 2.25i)T \)
19 \( 1 + (-18.9 - 0.100i)T \)
good5 \( 1 + (0.178 + 0.309i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (3.52 + 6.09i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (4.98 + 8.63i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 2.10iT - 169T^{2} \)
17 \( 1 + (-7.21 + 12.5i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 - 45.4T + 529T^{2} \)
29 \( 1 + (3.90 + 2.25i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-19.9 - 11.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 14.4iT - 1.36e3T^{2} \)
41 \( 1 + (-24.7 + 14.2i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + 47.8T + 1.84e3T^{2} \)
47 \( 1 + (0.891 - 1.54i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-24.6 + 14.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-10.1 + 5.83i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (17.5 - 30.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + 29.8iT - 4.48e3T^{2} \)
71 \( 1 + (74.5 + 43.0i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-2.28 + 3.95i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 - 66.5iT - 6.24e3T^{2} \)
83 \( 1 + (8.58 + 14.8i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-55.1 + 31.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 182. iT - 9.40e3T^{2} \)
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

−10.23364592941704572028754237564, −9.390539316655554223639840357817, −8.618597555159025374297901369942, −7.66766614853550259606315901485, −6.88593846898927231257305000759, −5.47203562619266486956862350651, −4.66412572089369795265062048343, −3.42905872062626403465569597000, −2.84397419438911490741766802668, −0.78205334478430769977356079905, 1.29216207700601546585972018281, 2.61942369251179638378630403868, 3.36834505371604703692498251863, 4.93663972060549415789541729601, 5.97447866363044747735971251831, 6.97053835293043348957627810245, 7.62736482038759972276442320765, 8.676825190353397451569033609209, 9.304219799820096046657537650549, 10.11469140216041512808178104371

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