Properties

Label 2-570-285.29-c1-0-19
Degree $2$
Conductor $570$
Sign $0.615 - 0.787i$
Analytic cond. $4.55147$
Root an. cond. $2.13341$
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  + (−0.984 + 0.173i)2-s + (−1.37 + 1.05i)3-s + (0.939 − 0.342i)4-s + (1.98 + 1.02i)5-s + (1.16 − 1.27i)6-s + (3.69 + 2.13i)7-s + (−0.866 + 0.5i)8-s + (0.770 − 2.89i)9-s + (−2.13 − 0.665i)10-s + (2.36 − 1.36i)11-s + (−0.929 + 1.46i)12-s + (2.74 + 2.30i)13-s + (−4.01 − 1.46i)14-s + (−3.81 + 0.689i)15-s + (0.766 − 0.642i)16-s + (−1.27 − 7.23i)17-s + ⋯
L(s)  = 1  + (−0.696 + 0.122i)2-s + (−0.792 + 0.609i)3-s + (0.469 − 0.171i)4-s + (0.888 + 0.458i)5-s + (0.477 − 0.521i)6-s + (1.39 + 0.806i)7-s + (−0.306 + 0.176i)8-s + (0.256 − 0.966i)9-s + (−0.675 − 0.210i)10-s + (0.712 − 0.411i)11-s + (−0.268 + 0.421i)12-s + (0.762 + 0.639i)13-s + (−1.07 − 0.390i)14-s + (−0.984 + 0.178i)15-s + (0.191 − 0.160i)16-s + (−0.309 − 1.75i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.615 - 0.787i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ 0.615 - 0.787i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07811 + 0.525797i\)
\(L(\frac12)\) \(\approx\) \(1.07811 + 0.525797i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.984 - 0.173i)T \)
3 \( 1 + (1.37 - 1.05i)T \)
5 \( 1 + (-1.98 - 1.02i)T \)
19 \( 1 + (-0.726 + 4.29i)T \)
good7 \( 1 + (-3.69 - 2.13i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.36 + 1.36i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.74 - 2.30i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.27 + 7.23i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (0.504 - 0.183i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.17 + 6.65i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-6.25 - 3.61i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.30T + 37T^{2} \)
41 \( 1 + (4.84 - 4.06i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-0.172 + 0.473i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.399 - 2.26i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-0.105 - 0.291i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (-1.70 - 9.66i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-1.45 + 0.530i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.976 - 5.53i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-2.79 - 1.01i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (8.13 + 9.69i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (9.01 + 10.7i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (7.26 - 12.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.27 + 6.10i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-0.308 - 1.75i)T + (-91.1 + 33.1i)T^{2} \)
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

−10.92305787708217153968510988113, −9.943214249701130274346730104654, −9.109681501953178089247417102804, −8.612333885101119824562698531228, −7.08767440525249639412806145727, −6.33485110312094902736685849153, −5.41637194719334237848000840619, −4.57634961806892719259148124398, −2.76151029553588992173255723072, −1.33392103089017093632656996184, 1.31578907532047751515786608129, 1.73661430586358951288052884288, 4.04577396715638868204858968420, 5.23285065374675465176964665909, 6.14722151399694390816803502964, 7.01490860258442343205404515608, 8.194810262899959449903106117865, 8.515052434146010185254697930255, 10.11009225598467079937429041086, 10.51281807944864759615593498789

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