↑

Properties

Label 2-684-171.11-c2-0-18
Degree $2$
Conductor $684$
Sign $0.643 + 0.765i$
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  + (−1.60 − 2.53i)3-s + 1.56i·5-s + (−4.59 + 7.96i)7-s + (−3.87 + 8.12i)9-s + (−18.3 − 10.6i)11-s + (0.163 − 0.282i)13-s + (3.96 − 2.50i)15-s + (22.8 + 13.2i)17-s + (8.51 − 16.9i)19-s + (27.5 − 1.07i)21-s + (−2.17 − 1.25i)23-s + 22.5·25-s + (26.8 − 3.15i)27-s − 52.9i·29-s + (−4.11 − 7.12i)31-s + ⋯
L(s)  = 1  + (−0.533 − 0.845i)3-s + 0.312i·5-s + (−0.656 + 1.13i)7-s + (−0.430 + 0.902i)9-s + (−1.67 − 0.964i)11-s + (0.0125 − 0.0217i)13-s + (0.264 − 0.166i)15-s + (1.34 + 0.776i)17-s + (0.448 − 0.893i)19-s + (1.31 − 0.0512i)21-s + (−0.0946 − 0.0546i)23-s + 0.902·25-s + (0.993 − 0.116i)27-s − 1.82i·29-s + (−0.132 − 0.229i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.643 + 0.765i$
Analytic conductor: \(18.6376\)
Root analytic conductor: \(4.31713\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1),\ 0.643 + 0.765i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.073146842\)
\(L(\frac12)\) \(\approx\) \(1.073146842\)
\(L(2)\) 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 \( 1 + (1.60 + 2.53i)T \)
19 \( 1 + (-8.51 + 16.9i)T \)
good5 \( 1 - 1.56iT - 25T^{2} \)
7 \( 1 + (4.59 - 7.96i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (18.3 + 10.6i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-0.163 + 0.282i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-22.8 - 13.2i)T + (144.5 + 250. i)T^{2} \)
23 \( 1 + (2.17 + 1.25i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 52.9iT - 841T^{2} \)
31 \( 1 + (4.11 + 7.12i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 30.2T + 1.36e3T^{2} \)
41 \( 1 - 10.9iT - 1.68e3T^{2} \)
43 \( 1 + (-30.7 - 53.3i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 - 41.4iT - 2.20e3T^{2} \)
53 \( 1 + (-54.4 + 31.4i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + 78.8iT - 3.48e3T^{2} \)
61 \( 1 + 57.7T + 3.72e3T^{2} \)
67 \( 1 + (-30.3 + 52.5i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-48.8 - 28.1i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-30.1 + 52.2i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (34.6 + 60.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-1.98 - 1.14i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + (-92.6 + 53.4i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-13.8 - 24.0i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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

−10.31175616050271691906895096294, −9.295536400479929676450112174000, −8.092114759262708747044020844190, −7.73723384459106403325935616474, −6.31910044051502810793743450447, −5.86254071507137774077256823312, −5.03711408260511013130579883159, −3.07889469162376482058038121045, −2.44681424268756114919579430326, −0.58929379744298661877476754861, 0.844894697363317559077278931326, 2.95787339552109557210216496056, 3.92042329026696509075591273083, 5.04053119025667560050862736237, 5.55772706586301868245189830003, 7.02796600465051727899202692982, 7.57687279291024632556005304977, 8.863340805355106629555900028930, 9.957174164258719087943582068541, 10.21961589987268443644750471299

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