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Properties

Label 2-513-171.164-c1-0-9
Degree $2$
Conductor $513$
Sign $0.981 + 0.193i$
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  − 1.24·2-s − 0.457·4-s + (3.19 + 1.84i)5-s + (1.96 − 3.39i)7-s + 3.05·8-s + (−3.97 − 2.29i)10-s + (−0.341 − 0.197i)11-s + 0.579i·13-s + (−2.43 + 4.22i)14-s − 2.87·16-s + (−1.72 + 0.997i)17-s + (0.232 − 4.35i)19-s + (−1.46 − 0.844i)20-s + (0.423 + 0.244i)22-s − 5.70i·23-s + ⋯
L(s)  = 1  − 0.878·2-s − 0.228·4-s + (1.43 + 0.825i)5-s + (0.741 − 1.28i)7-s + 1.07·8-s + (−1.25 − 0.725i)10-s + (−0.102 − 0.0594i)11-s + 0.160i·13-s + (−0.651 + 1.12i)14-s − 0.719·16-s + (−0.418 + 0.241i)17-s + (0.0533 − 0.998i)19-s + (−0.327 − 0.188i)20-s + (0.0903 + 0.0521i)22-s − 1.18i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.981 + 0.193i$
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: \((2,\ 513,\ (\ :1/2),\ 0.981 + 0.193i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13067 - 0.110638i\)
\(L(\frac12)\) \(\approx\) \(1.13067 - 0.110638i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.232 + 4.35i)T \)
good2 \( 1 + 1.24T + 2T^{2} \)
5 \( 1 + (-3.19 - 1.84i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.96 + 3.39i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.341 + 0.197i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.579iT - 13T^{2} \)
17 \( 1 + (1.72 - 0.997i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + 5.70iT - 23T^{2} \)
29 \( 1 + (-4.98 - 8.62i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.68 + 0.970i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.13iT - 37T^{2} \)
41 \( 1 + (1.97 - 3.42i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 + (0.00863 - 0.00498i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.06 + 7.03i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.541 - 0.937i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.15 + 5.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 - 9.85iT - 67T^{2} \)
71 \( 1 + (-2.79 - 4.84i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.876 - 1.51i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 11.0iT - 79T^{2} \)
83 \( 1 + (3.28 + 1.89i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.85 - 4.95i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.34iT - 97T^{2} \)
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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

−10.69540864623974993565376330918, −10.06706037909366681184497946921, −9.185484617069815463551620484759, −8.340667768702679150976988825833, −7.19353756545201118372693617226, −6.63417642479274409815946879979, −5.19273158297185527054904541656, −4.21990498730854578475527006025, −2.47036573682648297521946254352, −1.14068357647685091191528590148, 1.37365163122742792167515045998, 2.33978491780190311765631283193, 4.52067323377046176344484494823, 5.39981939398771503892351621384, 6.07337981128372228296243034500, 7.74463422051947888000816669048, 8.497954754870183693775626986843, 9.184240252407017062877236623803, 9.734537759849556771039044305785, 10.57192377060367598867122130503

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