↑

Properties

Label 2-513-171.49-c1-0-17
Degree $2$
Conductor $513$
Sign $0.0202 + 0.999i$
Analytic cond. $4.09632$
Root an. cond. $2.02393$
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  + 2.02·2-s + 2.08·4-s + (−2.09 − 3.62i)5-s + (−0.976 − 1.69i)7-s + 0.164·8-s + (−4.22 − 7.32i)10-s + (0.669 + 1.15i)11-s + 1.95·13-s + (−1.97 − 3.41i)14-s − 3.83·16-s + (3.34 − 5.78i)17-s + (4.11 − 1.44i)19-s + (−4.35 − 7.54i)20-s + (1.35 + 2.34i)22-s + 1.97·23-s + ⋯
L(s)  = 1  + 1.42·2-s + 1.04·4-s + (−0.936 − 1.62i)5-s + (−0.368 − 0.639i)7-s + 0.0583·8-s + (−1.33 − 2.31i)10-s + (0.201 + 0.349i)11-s + 0.541·13-s + (−0.527 − 0.912i)14-s − 0.957·16-s + (0.810 − 1.40i)17-s + (0.943 − 0.331i)19-s + (−0.974 − 1.68i)20-s + (0.288 + 0.499i)22-s + 0.411·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0202 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0202 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.0202 + 0.999i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (334, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ 0.0202 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65723 - 1.62405i\)
\(L(\frac12)\) \(\approx\) \(1.65723 - 1.62405i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 + (-4.11 + 1.44i)T \)
good2 \( 1 - 2.02T + 2T^{2} \)
5 \( 1 + (2.09 + 3.62i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.976 + 1.69i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.669 - 1.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.95T + 13T^{2} \)
17 \( 1 + (-3.34 + 5.78i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 - 1.97T + 23T^{2} \)
29 \( 1 + (2.95 - 5.11i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.385 - 0.668i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.26T + 37T^{2} \)
41 \( 1 + (-3.79 - 6.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 7.95T + 43T^{2} \)
47 \( 1 + (0.553 - 0.959i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.75 + 6.49i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.506 - 0.878i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.166 - 0.287i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 + (1.81 - 3.14i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.48 - 4.29i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 5.17T + 79T^{2} \)
83 \( 1 + (6.30 + 10.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.569 + 0.985i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.75T + 97T^{2} \)
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

−11.26311101521346878543949934562, −9.608977634805456295223429502273, −8.979656736847489069123653303806, −7.74942465078303519901623074344, −6.95970675724109506050300292273, −5.51343866077807351944755285313, −4.89455509070653240996893552608, −4.04990869857872322174752018672, −3.20041307224828997817933868480, −0.929626932781729639669806204438, 2.60045783426700151650716032344, 3.49660221929731939442986710679, 4.02717895699810790899091338910, 5.75064448379821908396246413438, 6.15881824157569069707997927686, 7.22189527719359969641714033509, 8.137752855143648396930562700528, 9.463335572983100266034650850921, 10.69965543620543851811365560432, 11.27497866724436043547610579189

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