↑

Properties

Label 2-414-9.7-c1-0-15
Degree $2$
Conductor $414$
Sign $0.0321 + 0.999i$
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 + (0.706 − 1.58i)3-s + (−0.499 − 0.866i)4-s + (1.63 + 2.83i)5-s + (−1.01 − 1.40i)6-s + (−0.135 + 0.235i)7-s − 0.999·8-s + (−2.00 − 2.23i)9-s + 3.27·10-s + (2.69 − 4.66i)11-s + (−1.72 + 0.178i)12-s + (−1.98 − 3.43i)13-s + (0.135 + 0.235i)14-s + (5.64 − 0.585i)15-s + (−0.5 + 0.866i)16-s + 2.00·17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.407 − 0.912i)3-s + (−0.249 − 0.433i)4-s + (0.732 + 1.26i)5-s + (−0.414 − 0.572i)6-s + (−0.0512 + 0.0888i)7-s − 0.353·8-s + (−0.667 − 0.744i)9-s + 1.03·10-s + (0.811 − 1.40i)11-s + (−0.497 + 0.0515i)12-s + (−0.550 − 0.954i)13-s + (0.0362 + 0.0628i)14-s + (1.45 − 0.151i)15-s + (−0.125 + 0.216i)16-s + 0.485·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0321 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0321 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $0.0321 + 0.999i$
Analytic conductor: \(3.30580\)
Root analytic conductor: \(1.81818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{414} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :1/2),\ 0.0321 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44542 - 1.39962i\)
\(L(\frac12)\) \(\approx\) \(1.44542 - 1.39962i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.706 + 1.58i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-1.63 - 2.83i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.135 - 0.235i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.69 + 4.66i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.98 + 3.43i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.00T + 17T^{2} \)
19 \( 1 - 4.41T + 19T^{2} \)
29 \( 1 + (1.19 - 2.06i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.43 - 7.67i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.59T + 37T^{2} \)
41 \( 1 + (2.24 + 3.88i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.07 - 8.78i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.31 - 7.47i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.510T + 53T^{2} \)
59 \( 1 + (6.19 + 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.68 - 8.11i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.47 - 12.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.94T + 71T^{2} \)
73 \( 1 + 5.62T + 73T^{2} \)
79 \( 1 + (-0.0179 + 0.0311i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.16 + 8.94i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + (-5.83 + 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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

−11.11194244006556846823811889358, −10.22042811114961085121860444372, −9.322179676316485174446167249985, −8.246718954067619948581018350162, −7.09748484477647243675025138061, −6.24264828858543431863708928833, −5.45348647438486006599742050849, −3.20077721683084625728568914072, −3.00626048520851179876570061752, −1.34148018563562391188365159611, 1.99653120250971840541128952589, 3.84749641651644465478239896975, 4.75721336900576751523282275171, 5.33420004722562666645350635894, 6.68624810753466998523812333541, 7.83083000546542213853354926336, 8.913982689618487740796842835065, 9.549248716558227839898371459408, 10.00132702947908899124013772624, 11.75722702559450527760659188133

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