↑

Properties

Label 2-323-323.189-c0-0-0
Degree $2$
Conductor $323$
Sign $0.739 - 0.673i$
Analytic cond. $0.161197$
Root an. cond. $0.401494$
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  + i·4-s + (−0.292 − 0.707i)5-s + (−0.707 + 1.70i)7-s + (0.707 − 0.707i)9-s + (1.70 + 0.707i)11-s − 16-s + (−0.707 − 0.707i)17-s + (−0.707 − 0.707i)19-s + (0.707 − 0.292i)20-s + (−0.707 − 0.292i)23-s + (0.292 − 0.292i)25-s + (−1.70 − 0.707i)28-s + 1.41·35-s + (0.707 + 0.707i)36-s + (−0.707 + 1.70i)44-s + (−0.707 − 0.292i)45-s + ⋯
L(s)  = 1  + i·4-s + (−0.292 − 0.707i)5-s + (−0.707 + 1.70i)7-s + (0.707 − 0.707i)9-s + (1.70 + 0.707i)11-s − 16-s + (−0.707 − 0.707i)17-s + (−0.707 − 0.707i)19-s + (0.707 − 0.292i)20-s + (−0.707 − 0.292i)23-s + (0.292 − 0.292i)25-s + (−1.70 − 0.707i)28-s + 1.41·35-s + (0.707 + 0.707i)36-s + (−0.707 + 1.70i)44-s + (−0.707 − 0.292i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(323\)    =    \(17 \cdot 19\)
Sign: $0.739 - 0.673i$
Analytic conductor: \(0.161197\)
Root analytic conductor: \(0.401494\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{323} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 323,\ (\ :0),\ 0.739 - 0.673i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7645858171\)
\(L(\frac12)\) \(\approx\) \(0.7645858171\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (0.707 + 0.707i)T \)
19 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 - iT^{2} \)
3 \( 1 + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
7 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
11 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
23 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.707 - 0.707i)T^{2} \)
31 \( 1 + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (0.707 + 0.707i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + (-1 - i)T + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.707 - 0.707i)T^{2} \)
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

−12.20774265948183318441034363786, −11.54406496258768609661432561789, −9.659186928242189859215073183287, −9.006102350092544960512653756913, −8.565165717675066509700182486016, −6.97269180544226260887988534788, −6.37513753945969566038180124782, −4.69804856552482613388626401216, −3.79334992864196046425507430452, −2.32698716554051800579488312078, 1.48930694970323161982086704140, 3.69023753944148863323579342458, 4.38408888456669127412157324807, 6.26483879699196864761742807216, 6.67702431485819547714370806204, 7.70657614155159074021461516890, 9.175817670973514467409025182130, 10.15158718920218278362639549372, 10.69660371920653128289326789427, 11.32938956404932440037378208205

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