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Properties

Label 2-495-55.32-c1-0-10
Degree $2$
Conductor $495$
Sign $0.180 - 0.983i$
Analytic cond. $3.95259$
Root an. cond. $1.98811$
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  + 2i·4-s + (1.65 + 1.5i)5-s + 3.31·11-s − 4·16-s + (−3 + 3.31i)20-s + (−6.15 + 6.15i)23-s + (0.5 + 4.97i)25-s + 9.94·31-s + (−8.47 − 8.47i)37-s + 6.63i·44-s + (2.68 + 2.68i)47-s + 7i·49-s + (9.63 − 9.63i)53-s + (5.5 + 4.97i)55-s − 3.31i·59-s + ⋯
L(s)  = 1  + i·4-s + (0.741 + 0.670i)5-s + 1.00·11-s − 16-s + (−0.670 + 0.741i)20-s + (−1.28 + 1.28i)23-s + (0.100 + 0.994i)25-s + 1.78·31-s + (−1.39 − 1.39i)37-s + 1.00i·44-s + (0.391 + 0.391i)47-s + i·49-s + (1.32 − 1.32i)53-s + (0.741 + 0.670i)55-s − 0.431i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.180 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.180 - 0.983i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.180 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22927 + 1.02396i\)
\(L(\frac12)\) \(\approx\) \(1.22927 + 1.02396i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-1.65 - 1.5i)T \)
11 \( 1 - 3.31T \)
good2 \( 1 - 2iT^{2} \)
7 \( 1 - 7iT^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (6.15 - 6.15i)T - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 9.94T + 31T^{2} \)
37 \( 1 + (8.47 + 8.47i)T + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (-2.68 - 2.68i)T + 47iT^{2} \)
53 \( 1 + (-9.63 + 9.63i)T - 53iT^{2} \)
59 \( 1 + 3.31iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (-1.52 - 1.52i)T + 67iT^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + 9iT - 89T^{2} \)
97 \( 1 + (13.4 + 13.4i)T + 97iT^{2} \)
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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

−11.29980799376889689886901262795, −10.16728987018481317876101452606, −9.384259518805435182276973509535, −8.474350129962873310597636566630, −7.42377249847246443719676892318, −6.65376058647720259342819785698, −5.68587382440119764255050413021, −4.17935379818873751739239735387, −3.24671931664923696691218546011, −1.97773268211154975998856887080, 1.05182651121211243436205258059, 2.27457280885841277135217566197, 4.20029597274579801027974039225, 5.10550727424362354083474764831, 6.14999519712168288598715564797, 6.69540845508282376977708910248, 8.356833277586075117190941220507, 9.023333709938057754468999631370, 10.02568575635188047763248188206, 10.37838596890461048907584965096

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