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Properties

Label 2-513-171.164-c1-0-4
Degree $2$
Conductor $513$
Sign $0.481 - 0.876i$
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  − 0.280·2-s − 1.92·4-s + (−2.62 − 1.51i)5-s + (0.223 − 0.387i)7-s + 1.09·8-s + (0.736 + 0.424i)10-s + (0.843 + 0.487i)11-s + 4.61i·13-s + (−0.0626 + 0.108i)14-s + 3.53·16-s + (−2.33 + 1.34i)17-s + (3.66 − 2.36i)19-s + (5.04 + 2.91i)20-s + (−0.236 − 0.136i)22-s + 4.58i·23-s + ⋯
L(s)  = 1  − 0.198·2-s − 0.960·4-s + (−1.17 − 0.678i)5-s + (0.0845 − 0.146i)7-s + 0.388·8-s + (0.232 + 0.134i)10-s + (0.254 + 0.146i)11-s + 1.27i·13-s + (−0.0167 + 0.0290i)14-s + 0.883·16-s + (−0.566 + 0.327i)17-s + (0.840 − 0.541i)19-s + (1.12 + 0.651i)20-s + (−0.0504 − 0.0290i)22-s + 0.955i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.481 - 0.876i$
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: \((2,\ 513,\ (\ :1/2),\ 0.481 - 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.560749 + 0.331841i\)
\(L(\frac12)\) \(\approx\) \(0.560749 + 0.331841i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.66 + 2.36i)T \)
good2 \( 1 + 0.280T + 2T^{2} \)
5 \( 1 + (2.62 + 1.51i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.223 + 0.387i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.843 - 0.487i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.61iT - 13T^{2} \)
17 \( 1 + (2.33 - 1.34i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 - 4.58iT - 23T^{2} \)
29 \( 1 + (-5.19 - 8.99i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.94 + 2.85i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.14iT - 37T^{2} \)
41 \( 1 + (0.215 - 0.372i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 4.20T + 43T^{2} \)
47 \( 1 + (5.33 - 3.07i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.52 + 7.83i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.35 - 5.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.01 + 10.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + 3.07iT - 67T^{2} \)
71 \( 1 + (-1.40 - 2.43i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.88 - 3.25i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 13.5iT - 79T^{2} \)
83 \( 1 + (0.972 + 0.561i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.27 + 5.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 11.2iT - 97T^{2} \)
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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

−11.22045265352255756884722615972, −9.944040350493056080995341261285, −9.109619775083466767012634382850, −8.493068396111857923631675986015, −7.64876238911907409142909809278, −6.63523842380844894711849405028, −4.98715410173721804601687776890, −4.45630713780792933953177563440, −3.48215067568179090765436875056, −1.21873949528632161058240551394, 0.51441202938287393487568646770, 2.93907705648260137896073722928, 3.93223503282829175294415779037, 4.89700646172325185363498739225, 6.12294307649952654617192177662, 7.40572546258150140477360433969, 8.082158663462151490872643994676, 8.779180342574853188899704559394, 10.00292250799901770362859528308, 10.58039687928905547991097242005

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