↑

Properties

Label 2-2646-9.4-c1-0-20
Degree $2$
Conductor $2646$
Sign $0.677 + 0.735i$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
Motivic weight $1$
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 + (−0.499 + 0.866i)4-s + (0.296 − 0.514i)5-s + 0.999·8-s − 0.593·10-s + (−0.296 − 0.514i)11-s + (−1.25 + 2.17i)13-s + (−0.5 − 0.866i)16-s + 2.92·17-s − 5.38·19-s + (0.296 + 0.514i)20-s + (−0.296 + 0.514i)22-s + (2.23 − 3.86i)23-s + (2.32 + 4.02i)25-s + 2.51·26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.132 − 0.229i)5-s + 0.353·8-s − 0.187·10-s + (−0.0894 − 0.154i)11-s + (−0.348 + 0.603i)13-s + (−0.125 − 0.216i)16-s + 0.708·17-s − 1.23·19-s + (0.0663 + 0.114i)20-s + (−0.0632 + 0.109i)22-s + (0.465 − 0.805i)23-s + (0.464 + 0.804i)25-s + 0.493·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.677 + 0.735i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (1765, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.677 + 0.735i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.400817747\)
\(L(\frac12)\) \(\approx\) \(1.400817747\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.296 + 0.514i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.296 + 0.514i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.25 - 2.17i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.92T + 17T^{2} \)
19 \( 1 + 5.38T + 19T^{2} \)
23 \( 1 + (-2.23 + 3.86i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.09 - 5.36i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.93 + 6.81i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-0.136 + 0.236i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.58 + 9.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.08 - 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.05T + 53T^{2} \)
59 \( 1 + (-4.32 + 7.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.32 - 5.75i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.956 + 1.65i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 + 7.91T + 73T^{2} \)
79 \( 1 + (-4.62 - 8.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.85 + 6.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + (-5.86 - 10.1i)T + (-48.5 + 84.0i)T^{2} \)
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

−8.760296895685320586108805011002, −8.299259057359489345403387671947, −7.26942498839023124193822791263, −6.60734438695029697341738954798, −5.55434090509451544806641502620, −4.69452551232584104454919559066, −3.90485605118626574547609901547, −2.84848826350845918595296649361, −1.97045152944271592003250405072, −0.76076830674428686830131236024, 0.815376185556146727469851579262, 2.20928941181920677413335479916, 3.23325572092627455370510861153, 4.39411818509083939274297372996, 5.17619647204596413148677058829, 5.99325111900113740231976761120, 6.74100240985282832047089824824, 7.39515490174509716265637396049, 8.328150414147929921741275199367, 8.648389979126376199141080048508

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