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Properties

Label 2-495-5.2-c2-0-45
Degree $2$
Conductor $495$
Sign $-0.852 + 0.523i$
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  + (0.364 + 0.364i)2-s − 3.73i·4-s + (0.0133 − 4.99i)5-s + (−2.00 − 2.00i)7-s + (2.82 − 2.82i)8-s + (1.82 − 1.81i)10-s − 3.31·11-s + (3.51 − 3.51i)13-s − 1.45i·14-s − 12.8·16-s + (8.30 + 8.30i)17-s − 19.7i·19-s + (−18.6 − 0.0498i)20-s + (−1.20 − 1.20i)22-s + (−2.95 + 2.95i)23-s + ⋯
L(s)  = 1  + (0.182 + 0.182i)2-s − 0.933i·4-s + (0.00266 − 0.999i)5-s + (−0.285 − 0.285i)7-s + (0.352 − 0.352i)8-s + (0.182 − 0.181i)10-s − 0.301·11-s + (0.270 − 0.270i)13-s − 0.104i·14-s − 0.804·16-s + (0.488 + 0.488i)17-s − 1.04i·19-s + (−0.933 − 0.00249i)20-s + (−0.0549 − 0.0549i)22-s + (−0.128 + 0.128i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 + 0.523i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.852 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.852 + 0.523i$
Analytic conductor: \(13.4877\)
Root analytic conductor: \(3.67257\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1),\ -0.852 + 0.523i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.312892863\)
\(L(\frac12)\) \(\approx\) \(1.312892863\)
\(L(2)\) 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 \( 1 + (-0.0133 + 4.99i)T \)
11 \( 1 + 3.31T \)
good2 \( 1 + (-0.364 - 0.364i)T + 4iT^{2} \)
7 \( 1 + (2.00 + 2.00i)T + 49iT^{2} \)
13 \( 1 + (-3.51 + 3.51i)T - 169iT^{2} \)
17 \( 1 + (-8.30 - 8.30i)T + 289iT^{2} \)
19 \( 1 + 19.7iT - 361T^{2} \)
23 \( 1 + (2.95 - 2.95i)T - 529iT^{2} \)
29 \( 1 - 33.7iT - 841T^{2} \)
31 \( 1 + 35.5T + 961T^{2} \)
37 \( 1 + (20.7 + 20.7i)T + 1.36e3iT^{2} \)
41 \( 1 - 10.5T + 1.68e3T^{2} \)
43 \( 1 + (25.7 - 25.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (8.30 + 8.30i)T + 2.20e3iT^{2} \)
53 \( 1 + (-29.2 + 29.2i)T - 2.80e3iT^{2} \)
59 \( 1 - 17.0iT - 3.48e3T^{2} \)
61 \( 1 - 2.32T + 3.72e3T^{2} \)
67 \( 1 + (55.1 + 55.1i)T + 4.48e3iT^{2} \)
71 \( 1 - 82.4T + 5.04e3T^{2} \)
73 \( 1 + (-57.3 + 57.3i)T - 5.32e3iT^{2} \)
79 \( 1 + 87.8iT - 6.24e3T^{2} \)
83 \( 1 + (-109. + 109. i)T - 6.88e3iT^{2} \)
89 \( 1 - 127. iT - 7.92e3T^{2} \)
97 \( 1 + (-49.8 - 49.8i)T + 9.40e3iT^{2} \)
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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

−10.37436268237969395081382811041, −9.443471656528386861464051885425, −8.745491747149143782355496909242, −7.58422142675195426636378433157, −6.54208371701108066718129592466, −5.49036904346792099704487606433, −4.87898113273115086415034005551, −3.62372451791834362587167650391, −1.76548582152868190488008943814, −0.48378504214937408697296531142, 2.19129022460752196090065474860, 3.21942631846687661852867090128, 4.04858133385526577196260655807, 5.53702875106197199807226379573, 6.60602346998168103720175101554, 7.51834934555956036709229398979, 8.236900815652869932339379670928, 9.397866286442221061288578044832, 10.31203022636933481330343699173, 11.22580843946972295459919388565

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