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Properties

Label 2-513-171.164-c1-0-16
Degree $2$
Conductor $513$
Sign $0.875 + 0.483i$
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  + 2.63·2-s + 4.93·4-s + (−2.90 − 1.67i)5-s + (1.12 − 1.94i)7-s + 7.73·8-s + (−7.65 − 4.41i)10-s + (3.25 + 1.88i)11-s − 1.84i·13-s + (2.95 − 5.12i)14-s + 10.4·16-s + (−1.61 + 0.933i)17-s + (−1.08 + 4.22i)19-s + (−14.3 − 8.27i)20-s + (8.57 + 4.95i)22-s + 3.58i·23-s + ⋯
L(s)  = 1  + 1.86·2-s + 2.46·4-s + (−1.29 − 0.750i)5-s + (0.424 − 0.735i)7-s + 2.73·8-s + (−2.41 − 1.39i)10-s + (0.981 + 0.566i)11-s − 0.512i·13-s + (0.790 − 1.36i)14-s + 2.62·16-s + (−0.392 + 0.226i)17-s + (−0.248 + 0.968i)19-s + (−3.20 − 1.85i)20-s + (1.82 + 1.05i)22-s + 0.748i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.875 + 0.483i$
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: \((2,\ 513,\ (\ :1/2),\ 0.875 + 0.483i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.59789 - 0.927799i\)
\(L(\frac12)\) \(\approx\) \(3.59789 - 0.927799i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.08 - 4.22i)T \)
good2 \( 1 - 2.63T + 2T^{2} \)
5 \( 1 + (2.90 + 1.67i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.12 + 1.94i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.25 - 1.88i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.84iT - 13T^{2} \)
17 \( 1 + (1.61 - 0.933i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 - 3.58iT - 23T^{2} \)
29 \( 1 + (0.128 + 0.221i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (9.25 - 5.34i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.06iT - 37T^{2} \)
41 \( 1 + (-0.777 + 1.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 3.27T + 43T^{2} \)
47 \( 1 + (-1.42 + 0.823i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.888 + 1.53i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.47 - 7.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.27 - 3.93i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 - 6.05iT - 67T^{2} \)
71 \( 1 + (0.0760 + 0.131i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.33 + 2.31i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 15.9iT - 79T^{2} \)
83 \( 1 + (-10.6 - 6.13i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.61 + 6.26i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.7iT - 97T^{2} \)
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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

−11.23866873974428758803600151776, −10.46357607426060515870161016878, −8.854819361177885980023982407458, −7.60842363195267828663636070536, −7.18404914852209880953402202190, −5.87978665477309422279301409457, −4.84628369553144985717195423349, −4.03834628230512236562829998310, −3.58430793141618107814242764723, −1.62372164732867065272688796251, 2.28335422676720293659156522508, 3.39802522063722295274382163476, 4.16851387665279828905137648910, 5.08305930987724478367864400786, 6.35624696322835654138152626563, 6.87939592144664392644484270099, 7.921573301573353646007460949486, 9.068831106900438088597625800501, 10.84110563955015345898802486130, 11.39482918332593353859180284133

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