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Properties

Label 2-495-15.14-c2-0-36
Degree $2$
Conductor $495$
Sign $0.676 + 0.736i$
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  + 3.68·2-s + 9.58·4-s + (−0.638 − 4.95i)5-s − 5.17i·7-s + 20.5·8-s + (−2.35 − 18.2i)10-s − 3.31i·11-s + 5.25i·13-s − 19.0i·14-s + 37.4·16-s − 20.0·17-s + 25.6·19-s + (−6.11 − 47.5i)20-s − 12.2i·22-s + 18.3·23-s + ⋯
L(s)  = 1  + 1.84·2-s + 2.39·4-s + (−0.127 − 0.991i)5-s − 0.738i·7-s + 2.57·8-s + (−0.235 − 1.82i)10-s − 0.301i·11-s + 0.404i·13-s − 1.36i·14-s + 2.34·16-s − 1.17·17-s + 1.35·19-s + (−0.305 − 2.37i)20-s − 0.555i·22-s + 0.795·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.676 + 0.736i$
Analytic conductor: \(13.4877\)
Root analytic conductor: \(3.67257\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1),\ 0.676 + 0.736i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(5.240510751\)
\(L(\frac12)\) \(\approx\) \(5.240510751\)
\(L(2)\) 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 \( 1 + (0.638 + 4.95i)T \)
11 \( 1 + 3.31iT \)
good2 \( 1 - 3.68T + 4T^{2} \)
7 \( 1 + 5.17iT - 49T^{2} \)
13 \( 1 - 5.25iT - 169T^{2} \)
17 \( 1 + 20.0T + 289T^{2} \)
19 \( 1 - 25.6T + 361T^{2} \)
23 \( 1 - 18.3T + 529T^{2} \)
29 \( 1 - 15.0iT - 841T^{2} \)
31 \( 1 - 3.38T + 961T^{2} \)
37 \( 1 - 16.8iT - 1.36e3T^{2} \)
41 \( 1 - 52.0iT - 1.68e3T^{2} \)
43 \( 1 - 25.1iT - 1.84e3T^{2} \)
47 \( 1 - 70.8T + 2.20e3T^{2} \)
53 \( 1 + 94.2T + 2.80e3T^{2} \)
59 \( 1 + 82.3iT - 3.48e3T^{2} \)
61 \( 1 - 60.9T + 3.72e3T^{2} \)
67 \( 1 - 127. iT - 4.48e3T^{2} \)
71 \( 1 - 56.7iT - 5.04e3T^{2} \)
73 \( 1 - 32.0iT - 5.32e3T^{2} \)
79 \( 1 + 101.T + 6.24e3T^{2} \)
83 \( 1 + 135.T + 6.88e3T^{2} \)
89 \( 1 + 147. iT - 7.92e3T^{2} \)
97 \( 1 - 71.8iT - 9.40e3T^{2} \)
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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

−11.29057021609776460043780964984, −9.933147146388555292640938315021, −8.737077592549312091813336992328, −7.49704234986916495915381493367, −6.71097177642158910332686195559, −5.60985871351497248648521261895, −4.74478808253232031804805834831, −4.08313097784917150394696822494, −2.93353887739965631820716400569, −1.32761029933472190928353226926, 2.22774042327853146843034293650, 3.01564545243259950705934095279, 4.05247117472088328353022364193, 5.20356350025401544864827019869, 5.97466202357392743352569307561, 6.91111371034889526099984922386, 7.59520060163154455559391864703, 9.140035935920651216943032831457, 10.46306231716328436055474558271, 11.16668693395019659126607585365

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