↑

Properties

Label 2-1352-104.3-c0-0-4
Degree $2$
Conductor $1352$
Sign $-0.434 + 0.900i$
Analytic cond. $0.674735$
Root an. cond. $0.821423$
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  + (−0.5 − 0.866i)2-s + (−0.623 − 1.07i)3-s + (−0.499 + 0.866i)4-s + (−0.623 + 1.07i)6-s + 0.999·8-s + (−0.277 + 0.480i)9-s + (0.900 + 1.56i)11-s + 1.24·12-s + (−0.5 − 0.866i)16-s + (0.900 − 1.56i)17-s + 0.554·18-s + (0.222 − 0.385i)19-s + (0.900 − 1.56i)22-s + (−0.623 − 1.07i)24-s + 25-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.623 − 1.07i)3-s + (−0.499 + 0.866i)4-s + (−0.623 + 1.07i)6-s + 0.999·8-s + (−0.277 + 0.480i)9-s + (0.900 + 1.56i)11-s + 1.24·12-s + (−0.5 − 0.866i)16-s + (0.900 − 1.56i)17-s + 0.554·18-s + (0.222 − 0.385i)19-s + (0.900 − 1.56i)22-s + (−0.623 − 1.07i)24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1352\)    =    \(2^{3} \cdot 13^{2}\)
Sign: $-0.434 + 0.900i$
Analytic conductor: \(0.674735\)
Root analytic conductor: \(0.821423\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1352} (315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1352,\ (\ :0),\ -0.434 + 0.900i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7029015698\)
\(L(\frac12)\) \(\approx\) \(0.7029015698\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 \)
good3 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 - 1.24T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 0.445T + T^{2} \)
89 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
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

−9.562072990706210042532653976573, −8.992907602711396569926917545839, −7.70866187122302152674829801427, −7.21884964484378095173018329355, −6.60766069517912428430611762558, −5.22311931300632149245700822172, −4.39956048042019784614161705636, −3.09849765306649776466332060840, −1.94499994786704589834429463247, −0.962837177863237268478921396996, 1.25534806810027383574110878426, 3.47477392196778424393563205161, 4.23027900361961567929050465010, 5.30090434681738521831494983969, 5.92536972279939553994150028393, 6.52505804788820968077692016181, 7.80895520632920861812672067410, 8.490624270980006470936455222649, 9.250986760581378643736395632644, 9.969197112230238799696574757979

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