↑

Properties

Label 2-414-9.4-c1-0-6
Degree $2$
Conductor $414$
Sign $0.377 - 0.926i$
Analytic cond. $3.30580$
Root an. cond. $1.81818$
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 + (−0.416 − 1.68i)3-s + (−0.499 + 0.866i)4-s + (−0.229 + 0.398i)5-s + (1.24 − 1.20i)6-s + (2.38 + 4.12i)7-s − 0.999·8-s + (−2.65 + 1.40i)9-s − 0.459·10-s + (0.173 + 0.300i)11-s + (1.66 + 0.480i)12-s + (−0.590 + 1.02i)13-s + (−2.38 + 4.12i)14-s + (0.765 + 0.220i)15-s + (−0.5 − 0.866i)16-s + 3.30·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.240 − 0.970i)3-s + (−0.249 + 0.433i)4-s + (−0.102 + 0.178i)5-s + (0.509 − 0.490i)6-s + (0.900 + 1.56i)7-s − 0.353·8-s + (−0.884 + 0.466i)9-s − 0.145·10-s + (0.0523 + 0.0907i)11-s + (0.480 + 0.138i)12-s + (−0.163 + 0.283i)13-s + (−0.636 + 1.10i)14-s + (0.197 + 0.0570i)15-s + (−0.125 − 0.216i)16-s + 0.801·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.377 - 0.926i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.377 - 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25116 + 0.841066i\)
\(L(\frac12)\) \(\approx\) \(1.25116 + 0.841066i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.416 + 1.68i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.229 - 0.398i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.38 - 4.12i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.173 - 0.300i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.590 - 1.02i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.30T + 17T^{2} \)
19 \( 1 - 3.47T + 19T^{2} \)
29 \( 1 + (-1.32 - 2.29i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.73 - 6.47i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.82T + 37T^{2} \)
41 \( 1 + (-5.04 + 8.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.09 + 8.82i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.49 + 6.05i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.836T + 53T^{2} \)
59 \( 1 + (-2.79 + 4.84i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.990 + 1.71i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.85 + 6.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.1T + 71T^{2} \)
73 \( 1 - 5.28T + 73T^{2} \)
79 \( 1 + (-0.432 - 0.749i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.69 + 8.12i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.11T + 89T^{2} \)
97 \( 1 + (-8.80 - 15.2i)T + (-48.5 + 84.0i)T^{2} \)
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

−11.88103317430738829627567940240, −10.76778782039043432544906949633, −9.130099385557310017000581786191, −8.491637405267573724659832642605, −7.54622913053417984466619439726, −6.75685568479837742309701372603, −5.47790434035407556100403007356, −5.19344228008474607745513872253, −3.17977067252652157861186018698, −1.83527039919616539746579362590, 1.00149511884867052719760985515, 3.12844248591848755962284186718, 4.21703272171057956295945425855, 4.81814380430291681744136296059, 5.94953934057108604660216961593, 7.45826566445531935691673048049, 8.362091507403022398640144832126, 9.746683528145422548784748576055, 10.15006515558984351615644441673, 11.19412456032201631772091496521

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