↑

Properties

Label 2-572-572.519-c1-0-29
Degree $2$
Conductor $572$
Sign $0.267 - 0.963i$
Analytic cond. $4.56744$
Root an. cond. $2.13715$
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  + (1.34 + 0.437i)2-s + (1.61 + 1.17i)4-s + (−3.10 + 4.27i)7-s + (1.66 + 2.28i)8-s + (0.927 − 2.85i)9-s + (3.30 − 0.217i)11-s + (3.42 + 1.11i)13-s + (−6.04 + 4.39i)14-s + (1.23 + 3.80i)16-s + (−5.73 + 1.86i)17-s + (2.49 − 3.43i)18-s + (1.60 + 2.20i)19-s + (4.54 + 1.15i)22-s + (−4.04 + 2.93i)25-s + (4.12 + 2.99i)26-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + (−1.17 + 1.61i)7-s + (0.587 + 0.809i)8-s + (0.309 − 0.951i)9-s + (0.997 − 0.0656i)11-s + (0.951 + 0.309i)13-s + (−1.61 + 1.17i)14-s + (0.309 + 0.951i)16-s + (−1.39 + 0.452i)17-s + (0.587 − 0.809i)18-s + (0.367 + 0.505i)19-s + (0.969 + 0.245i)22-s + (−0.809 + 0.587i)25-s + (0.809 + 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.267 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.267 - 0.963i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ 0.267 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96589 + 1.49497i\)
\(L(\frac12)\) \(\approx\) \(1.96589 + 1.49497i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 - 0.437i)T \)
11 \( 1 + (-3.30 + 0.217i)T \)
13 \( 1 + (-3.42 - 1.11i)T \)
good3 \( 1 + (-0.927 + 2.85i)T^{2} \)
5 \( 1 + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (3.10 - 4.27i)T + (-2.16 - 6.65i)T^{2} \)
17 \( 1 + (5.73 - 1.86i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.60 - 2.20i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-6.19 + 8.52i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.523 - 1.61i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-8.51 + 6.18i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.11 + 9.60i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (10.6 + 7.76i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-7.74 + 2.51i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 5.09T + 67T^{2} \)
71 \( 1 + (0.617 + 1.90i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (16.1 - 5.25i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (78.4 + 57.0i)T^{2} \)
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

−11.41865023191250080315475334579, −9.905338231327492233529577655359, −9.048028431377130009419505092757, −8.400330427293186967588284037339, −6.72633826422255343574565242861, −6.35423909710787469669343718911, −5.66362894157888913166787216784, −4.10582474890221271061372143589, −3.40067371515296877251138985087, −2.08947911116533873574367593168, 1.15006971032326475047628089855, 2.85615679774785083395335852790, 3.99480474638545008966487567300, 4.54514725943303376405733888369, 6.04838596662943098611396584048, 6.84662871998369623145767724781, 7.41562836099327665654812884875, 8.978746039817371677279377454418, 10.04102437022390960390231600076, 10.69241222997866708891602706706

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