↑

Properties

Label 2-684-684.463-c0-0-1
Degree $2$
Conductor $684$
Sign $-0.466 - 0.884i$
Analytic cond. $0.341360$
Root an. cond. $0.584260$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s − 5-s + (−0.866 + 0.499i)6-s + (−0.866 + 0.5i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.866 + 0.5i)11-s + 0.999i·12-s + 0.999i·14-s + (0.866 + 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + 0.999·18-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s − 5-s + (−0.866 + 0.499i)6-s + (−0.866 + 0.5i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.866 + 0.5i)11-s + 0.999i·12-s + 0.999i·14-s + (0.866 + 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + 0.999·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $-0.466 - 0.884i$
Analytic conductor: \(0.341360\)
Root analytic conductor: \(0.584260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :0),\ -0.466 - 0.884i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1436142367\)
\(L(\frac12)\) \(\approx\) \(0.1436142367\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + iT \)
good5 \( 1 + T + T^{2} \)
7 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + iT - T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + iT - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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

−10.36266841036622250365391070258, −9.549884854456694370205484007711, −8.442727012412790002931758451125, −7.24329650451570938986850129742, −6.49459295389384189361523738053, −5.29724749959638334398145852492, −4.67744483176066983515949167751, −3.35592013392333426669943827053, −2.22063264703717049269575553726, −0.14159959331513986664560243801, 3.38435382604974684747376353806, 3.95055533778415243390417073924, 4.96432877109877564290118389346, 6.00720573302143861580785247622, 6.64226700375609267925308166301, 7.70362909059043334043758431296, 8.372936840730853709834206602501, 9.625933627288850907340810982741, 10.41972937334671750496526943363, 11.39638945013397224206419560484

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