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Properties

Label 2-572-13.4-c1-0-2
Degree $2$
Conductor $572$
Sign $-0.796 - 0.604i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.308 + 0.535i)3-s + 3.59i·5-s + (−1.27 + 0.734i)7-s + (1.30 + 2.26i)9-s + (−0.866 − 0.5i)11-s + (3.34 − 1.35i)13-s + (−1.92 − 1.10i)15-s + (−3.71 − 6.43i)17-s + (−5.72 + 3.30i)19-s − 0.907i·21-s + (−3.07 + 5.33i)23-s − 7.90·25-s − 3.47·27-s + (1.37 − 2.38i)29-s + 4.07i·31-s + ⋯
L(s)  = 1  + (−0.178 + 0.308i)3-s + 1.60i·5-s + (−0.480 + 0.277i)7-s + (0.436 + 0.755i)9-s + (−0.261 − 0.150i)11-s + (0.926 − 0.375i)13-s + (−0.496 − 0.286i)15-s + (−0.900 − 1.56i)17-s + (−1.31 + 0.758i)19-s − 0.198i·21-s + (−0.641 + 1.11i)23-s − 1.58·25-s − 0.668·27-s + (0.255 − 0.442i)29-s + 0.731i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.796 - 0.604i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ -0.796 - 0.604i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.326998 + 0.971967i\)
\(L(\frac12)\) \(\approx\) \(0.326998 + 0.971967i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-3.34 + 1.35i)T \)
good3 \( 1 + (0.308 - 0.535i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 3.59iT - 5T^{2} \)
7 \( 1 + (1.27 - 0.734i)T + (3.5 - 6.06i)T^{2} \)
17 \( 1 + (3.71 + 6.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.72 - 3.30i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.07 - 5.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.37 + 2.38i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.07iT - 31T^{2} \)
37 \( 1 + (-9.03 - 5.21i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.15 - 2.97i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.23 - 2.13i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.6iT - 47T^{2} \)
53 \( 1 + 4.18T + 53T^{2} \)
59 \( 1 + (-1.01 + 0.587i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.509 + 0.882i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.14 - 3.54i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.15 + 0.664i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 10.6iT - 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + 1.48iT - 83T^{2} \)
89 \( 1 + (-7.53 - 4.34i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.07 - 1.20i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.10505050697122607042202644289, −10.23659148822109083949241452823, −9.672172954513454446858330634649, −8.329999833084564010877064650063, −7.42039507958606070479577417635, −6.51969342144278495440880936229, −5.78682671908653462585916493768, −4.40316986942774356057876515459, −3.24678716609960454991647261111, −2.27966808145664325630457273135, 0.58430516239409014550954758385, 1.95128338932958382737550453132, 4.05266825760238996376017469266, 4.44046612237114954293856605423, 6.05551125942855956244118861145, 6.46245201981431221068175352335, 7.88737174262443788223417937572, 8.769942122059383982370915230050, 9.234868283247006933256602470581, 10.40750375943162063890623945353

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