↑

Properties

Label 2-1407-1407.1259-c0-0-1
Degree $2$
Conductor $1407$
Sign $0.987 - 0.156i$
Analytic cond. $0.702184$
Root an. cond. $0.837964$
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  + (0.841 + 0.540i)3-s + (0.654 − 0.755i)4-s + (0.959 + 0.281i)7-s + (0.415 + 0.909i)9-s + (0.959 − 0.281i)12-s + (−0.797 + 0.234i)13-s + (−0.142 − 0.989i)16-s + (−0.983 − 0.449i)19-s + (0.654 + 0.755i)21-s + (−0.959 + 0.281i)25-s + (−0.142 + 0.989i)27-s + (0.841 − 0.540i)28-s + (−0.273 − 0.0801i)31-s + (0.959 + 0.281i)36-s − 1.91·37-s + ⋯
L(s)  = 1  + (0.841 + 0.540i)3-s + (0.654 − 0.755i)4-s + (0.959 + 0.281i)7-s + (0.415 + 0.909i)9-s + (0.959 − 0.281i)12-s + (−0.797 + 0.234i)13-s + (−0.142 − 0.989i)16-s + (−0.983 − 0.449i)19-s + (0.654 + 0.755i)21-s + (−0.959 + 0.281i)25-s + (−0.142 + 0.989i)27-s + (0.841 − 0.540i)28-s + (−0.273 − 0.0801i)31-s + (0.959 + 0.281i)36-s − 1.91·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.987 - 0.156i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.987 - 0.156i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.711916039\)
\(L(\frac12)\) \(\approx\) \(1.711916039\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.841 - 0.540i)T \)
7 \( 1 + (-0.959 - 0.281i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
good2 \( 1 + (-0.654 + 0.755i)T^{2} \)
5 \( 1 + (0.959 - 0.281i)T^{2} \)
11 \( 1 + (-0.959 + 0.281i)T^{2} \)
13 \( 1 + (0.797 - 0.234i)T + (0.841 - 0.540i)T^{2} \)
17 \( 1 + (-0.142 + 0.989i)T^{2} \)
19 \( 1 + (0.983 + 0.449i)T + (0.654 + 0.755i)T^{2} \)
23 \( 1 + (-0.415 - 0.909i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.273 + 0.0801i)T + (0.841 + 0.540i)T^{2} \)
37 \( 1 + 1.91T + T^{2} \)
41 \( 1 + (0.142 - 0.989i)T^{2} \)
43 \( 1 + (-0.425 + 0.368i)T + (0.142 - 0.989i)T^{2} \)
47 \( 1 + (0.415 + 0.909i)T^{2} \)
53 \( 1 + (-0.142 - 0.989i)T^{2} \)
59 \( 1 + (0.841 + 0.540i)T^{2} \)
61 \( 1 + (-0.239 + 1.66i)T + (-0.959 - 0.281i)T^{2} \)
71 \( 1 + (0.142 + 0.989i)T^{2} \)
73 \( 1 + (-1.95 - 0.281i)T + (0.959 + 0.281i)T^{2} \)
79 \( 1 + (0.158 + 0.540i)T + (-0.841 + 0.540i)T^{2} \)
83 \( 1 + (-0.959 + 0.281i)T^{2} \)
89 \( 1 + (0.415 - 0.909i)T^{2} \)
97 \( 1 + 1.68T + T^{2} \)
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

−9.731790987765777341482406624733, −9.065468806126492003241117960480, −8.201867685992309247955855425749, −7.44388289436317599313936362426, −6.60850076109733929302304532750, −5.36799719017299308456128230294, −4.85504551592230790595925916667, −3.74646956405012640712037958919, −2.38469994896375112167751255479, −1.84942536810740582021549014768, 1.73655498633745415353986563456, 2.45469640836783267899158679907, 3.59888200039990674933811307504, 4.38701754056460220323051705284, 5.71723009672069557709845939599, 6.87399582489922529460319899410, 7.33656502616934254002319841158, 8.228266304990862858459860869367, 8.471821146815490396783116147302, 9.663642927582907611196020751227

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