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Properties

Label 2-495-3.2-c2-0-0
Degree $2$
Conductor $495$
Sign $0.577 + 0.816i$
Analytic cond. $13.4877$
Root an. cond. $3.67257$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.01i·2-s − 5.07·4-s + 2.23i·5-s − 9.19·7-s − 3.23i·8-s − 6.73·10-s + 3.31i·11-s − 3.78·13-s − 27.7i·14-s − 10.5·16-s − 6.22i·17-s + 11.1·19-s − 11.3i·20-s − 9.99·22-s − 6.99i·23-s + ⋯
L(s)  = 1  + 1.50i·2-s − 1.26·4-s + 0.447i·5-s − 1.31·7-s − 0.404i·8-s − 0.673·10-s + 0.301i·11-s − 0.291·13-s − 1.97i·14-s − 0.659·16-s − 0.366i·17-s + 0.588·19-s − 0.567i·20-s − 0.454·22-s − 0.304i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(13.4877\)
Root analytic conductor: \(3.67257\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (386, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1),\ 0.577 + 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.06353257287\)
\(L(\frac12)\) \(\approx\) \(0.06353257287\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - 2.23iT \)
11 \( 1 - 3.31iT \)
good2 \( 1 - 3.01iT - 4T^{2} \)
7 \( 1 + 9.19T + 49T^{2} \)
13 \( 1 + 3.78T + 169T^{2} \)
17 \( 1 + 6.22iT - 289T^{2} \)
19 \( 1 - 11.1T + 361T^{2} \)
23 \( 1 + 6.99iT - 529T^{2} \)
29 \( 1 + 34.5iT - 841T^{2} \)
31 \( 1 + 13.4T + 961T^{2} \)
37 \( 1 + 18.4T + 1.36e3T^{2} \)
41 \( 1 + 36.9iT - 1.68e3T^{2} \)
43 \( 1 + 16.3T + 1.84e3T^{2} \)
47 \( 1 + 63.9iT - 2.20e3T^{2} \)
53 \( 1 - 82.8iT - 2.80e3T^{2} \)
59 \( 1 + 11.2iT - 3.48e3T^{2} \)
61 \( 1 - 33.3T + 3.72e3T^{2} \)
67 \( 1 + 99.5T + 4.48e3T^{2} \)
71 \( 1 - 93.0iT - 5.04e3T^{2} \)
73 \( 1 + 69.4T + 5.32e3T^{2} \)
79 \( 1 + 9.29T + 6.24e3T^{2} \)
83 \( 1 - 137. iT - 6.88e3T^{2} \)
89 \( 1 - 12.5iT - 7.92e3T^{2} \)
97 \( 1 + 170.T + 9.40e3T^{2} \)
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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

−11.54148860390992289690573677230, −10.26157592927716779708588796562, −9.539062119370254926011846748425, −8.670983211030465181033550190919, −7.50227683273558414879865967503, −6.95693032006085889541591447603, −6.15074246310141154791491330003, −5.29570490577988203035450552531, −3.97874715395951138408180896103, −2.63375930799266300271409949779, 0.02538620928784173263280311784, 1.46725249943782337171563536566, 2.94688670235035225681639757433, 3.62096331249309130719154277393, 4.87056325208189999295761214703, 6.13687920231818665983476220521, 7.23756633804805961467625025011, 8.648728450625248715739776667610, 9.442191601104645917380478065282, 10.01740337789400986672556436151

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