↑

Properties

Label 2-513-171.49-c1-0-4
Degree $2$
Conductor $513$
Sign $0.983 + 0.180i$
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  − 1.77·2-s + 1.16·4-s + (−0.639 − 1.10i)5-s + (0.657 + 1.13i)7-s + 1.49·8-s + (1.13 + 1.97i)10-s + (0.130 + 0.225i)11-s − 1.86·13-s + (−1.16 − 2.02i)14-s − 4.97·16-s + (0.0508 − 0.0880i)17-s + (3.11 + 3.05i)19-s + (−0.742 − 1.28i)20-s + (−0.231 − 0.400i)22-s + 1.22·23-s + ⋯
L(s)  = 1  − 1.25·2-s + 0.580·4-s + (−0.286 − 0.495i)5-s + (0.248 + 0.430i)7-s + 0.527·8-s + (0.359 + 0.623i)10-s + (0.0392 + 0.0679i)11-s − 0.518·13-s + (−0.312 − 0.541i)14-s − 1.24·16-s + (0.0123 − 0.0213i)17-s + (0.713 + 0.700i)19-s + (−0.166 − 0.287i)20-s + (−0.0493 − 0.0854i)22-s + 0.255·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.983 + 0.180i$
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: \((2,\ 513,\ (\ :1/2),\ 0.983 + 0.180i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.686560 - 0.0625600i\)
\(L(\frac12)\) \(\approx\) \(0.686560 - 0.0625600i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.11 - 3.05i)T \)
good2 \( 1 + 1.77T + 2T^{2} \)
5 \( 1 + (0.639 + 1.10i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.657 - 1.13i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.130 - 0.225i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.86T + 13T^{2} \)
17 \( 1 + (-0.0508 + 0.0880i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 - 1.22T + 23T^{2} \)
29 \( 1 + (-3.26 + 5.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.617 - 1.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.59T + 37T^{2} \)
41 \( 1 + (-4.10 - 7.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 3.07T + 43T^{2} \)
47 \( 1 + (-0.790 + 1.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.59 - 4.50i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.01 - 6.94i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.06 + 12.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + 0.781T + 67T^{2} \)
71 \( 1 + (-8.19 + 14.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.397 - 0.687i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 + (-3.03 - 5.25i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.75 + 9.96i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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

−10.64399624113546143700904366724, −9.741213084047757604327096237082, −9.167761792865202385128657179209, −8.143520952968380257271369742546, −7.76027701622564818163964782871, −6.51931370235835880508930984037, −5.19693788096902042865054833781, −4.21190960023638067922534350798, −2.42718495632203441123738143291, −0.910347951386833501945419847220, 0.966751378020889750445119878552, 2.64504552609779296975854157781, 4.13042167786804648097596751465, 5.29684264634589070547810348638, 6.92653871631865886614221518841, 7.35152722995436517851029059313, 8.290856200059924957075049937173, 9.209168466275212518764609923449, 9.906360722236755315176263221449, 10.89613712692215099912593539813

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