↑

Properties

Label 2-322-7.2-c1-0-11
Degree $2$
Conductor $322$
Sign $0.307 + 0.951i$
Analytic cond. $2.57118$
Root an. cond. $1.60349$
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 + (0.319 + 0.553i)3-s + (−0.499 − 0.866i)4-s + (0.268 − 0.464i)5-s + 0.639·6-s + (0.716 − 2.54i)7-s − 0.999·8-s + (1.29 − 2.24i)9-s + (−0.268 − 0.464i)10-s + (2.07 + 3.60i)11-s + (0.319 − 0.553i)12-s − 2.41·13-s + (−1.84 − 1.89i)14-s + 0.342·15-s + (−0.5 + 0.866i)16-s + (−1.23 − 2.13i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.184 + 0.319i)3-s + (−0.249 − 0.433i)4-s + (0.119 − 0.207i)5-s + 0.260·6-s + (0.270 − 0.962i)7-s − 0.353·8-s + (0.431 − 0.748i)9-s + (−0.0848 − 0.146i)10-s + (0.626 + 1.08i)11-s + (0.0922 − 0.159i)12-s − 0.670·13-s + (−0.493 − 0.506i)14-s + 0.0885·15-s + (−0.125 + 0.216i)16-s + (−0.298 − 0.517i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.307 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.307 + 0.951i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.307 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37803 - 1.00289i\)
\(L(\frac12)\) \(\approx\) \(1.37803 - 1.00289i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.716 + 2.54i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (-0.319 - 0.553i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.268 + 0.464i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.07 - 3.60i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.41T + 13T^{2} \)
17 \( 1 + (1.23 + 2.13i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.42 + 5.93i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 1.01T + 29T^{2} \)
31 \( 1 + (0.560 + 0.970i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.62 - 9.74i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.65T + 41T^{2} \)
43 \( 1 + 3.50T + 43T^{2} \)
47 \( 1 + (4.86 - 8.42i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.44 - 11.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.45 + 2.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.23 - 12.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.45 - 2.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.21T + 71T^{2} \)
73 \( 1 + (2.59 + 4.49i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.32 + 9.22i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.4T + 83T^{2} \)
89 \( 1 + (1.85 - 3.20i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14.7T + 97T^{2} \)
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

−11.55139990084598471801550288725, −10.46023288806710443437165350038, −9.613134479171847226374472098749, −9.099409810794572725963812827670, −7.40305846459524601913374325311, −6.71736005204170625971217332543, −4.92300999803755893408245538827, −4.35485382163030707683636724915, −3.06088359487307738566057509435, −1.28319702701536308953631817514, 2.08469780738566910983275004200, 3.56278619369183984374335116871, 5.02879610486779812112847419202, 5.92118847574849756639720663094, 6.92219587238348008575470275928, 8.069595087961471376379259064997, 8.636339146994676066609943236241, 9.856135762555547287280472800929, 10.98354089847636511349349048434, 12.08604264180359481753487101025

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