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Properties

Label 2-513-171.164-c1-0-1
Degree $2$
Conductor $513$
Sign $-0.877 - 0.479i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.808·2-s − 1.34·4-s + (−1.00 − 0.578i)5-s + (−0.183 + 0.316i)7-s − 2.70·8-s + (−0.810 − 0.467i)10-s + (−1.84 − 1.06i)11-s + 5.93i·13-s + (−0.147 + 0.256i)14-s + 0.507·16-s + (−5.82 + 3.36i)17-s + (−1.77 + 3.97i)19-s + (1.35 + 0.779i)20-s + (−1.49 − 0.861i)22-s − 0.458i·23-s + ⋯
L(s)  = 1  + 0.571·2-s − 0.673·4-s + (−0.448 − 0.258i)5-s + (−0.0691 + 0.119i)7-s − 0.956·8-s + (−0.256 − 0.147i)10-s + (−0.556 − 0.321i)11-s + 1.64i·13-s + (−0.0395 + 0.0684i)14-s + 0.126·16-s + (−1.41 + 0.816i)17-s + (−0.407 + 0.913i)19-s + (0.301 + 0.174i)20-s + (−0.317 − 0.183i)22-s − 0.0956i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 - 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $-0.877 - 0.479i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (278, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ -0.877 - 0.479i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0874852 + 0.342864i\)
\(L(\frac12)\) \(\approx\) \(0.0874852 + 0.342864i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.77 - 3.97i)T \)
good2 \( 1 - 0.808T + 2T^{2} \)
5 \( 1 + (1.00 + 0.578i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.183 - 0.316i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.84 + 1.06i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.93iT - 13T^{2} \)
17 \( 1 + (5.82 - 3.36i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + 0.458iT - 23T^{2} \)
29 \( 1 + (4.29 + 7.44i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.53 - 2.61i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.13iT - 37T^{2} \)
41 \( 1 + (-0.345 + 0.598i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 4.70T + 43T^{2} \)
47 \( 1 + (-1.27 + 0.734i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.73 + 3.00i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.33 - 7.50i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.81 - 8.34i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + 8.23iT - 67T^{2} \)
71 \( 1 + (-2.74 - 4.75i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.46 + 4.26i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 5.65iT - 79T^{2} \)
83 \( 1 + (1.26 + 0.731i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.45 + 9.44i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.313iT - 97T^{2} \)
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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

−11.43518455356358487569975503539, −10.45939335205193942317379389308, −9.250964151707967288902627378109, −8.738812880525373471817378151325, −7.77467178608146828770884112572, −6.44635893862705234452366219011, −5.63742495799338604919511691932, −4.25923834849519036057174251095, −4.01245746841492617311240243180, −2.21302378620636673361064811286, 0.16806081711569842201870303121, 2.70699843771929248794010203399, 3.71014873492874405423087634966, 4.85484357081804229501011114129, 5.53187126570232081295133782104, 6.87346826296275329696724912917, 7.77039833291867882358698666500, 8.777138496600348622030350495464, 9.573484071131406490268865023440, 10.69030760175298760727066130698

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