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Properties

Label 2-495-11.5-c1-0-18
Degree $2$
Conductor $495$
Sign $-0.925 - 0.378i$
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  + (−0.633 − 1.94i)2-s + (−1.77 + 1.29i)4-s + (0.309 − 0.951i)5-s + (0.477 − 0.346i)7-s + (0.329 + 0.239i)8-s − 2.04·10-s + (−0.933 − 3.18i)11-s + (0.465 + 1.43i)13-s + (−0.977 − 0.710i)14-s + (−1.10 + 3.38i)16-s + (1.52 − 4.67i)17-s + (−5.51 − 4.00i)19-s + (0.679 + 2.09i)20-s + (−5.61 + 3.83i)22-s − 0.0822·23-s + ⋯
L(s)  = 1  + (−0.447 − 1.37i)2-s + (−0.889 + 0.646i)4-s + (0.138 − 0.425i)5-s + (0.180 − 0.131i)7-s + (0.116 + 0.0845i)8-s − 0.647·10-s + (−0.281 − 0.959i)11-s + (0.128 + 0.396i)13-s + (−0.261 − 0.189i)14-s + (−0.275 + 0.847i)16-s + (0.368 − 1.13i)17-s + (−1.26 − 0.919i)19-s + (0.151 + 0.467i)20-s + (−1.19 + 0.817i)22-s − 0.0171·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.925 - 0.378i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ -0.925 - 0.378i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.160741 + 0.816658i\)
\(L(\frac12)\) \(\approx\) \(0.160741 + 0.816658i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (0.933 + 3.18i)T \)
good2 \( 1 + (0.633 + 1.94i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + (-0.477 + 0.346i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.465 - 1.43i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.52 + 4.67i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.51 + 4.00i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 0.0822T + 23T^{2} \)
29 \( 1 + (6.81 - 4.95i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.611 + 1.88i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.89 + 2.10i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.29 + 2.39i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 4.86T + 43T^{2} \)
47 \( 1 + (4.67 + 3.39i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.09 - 6.44i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-4.51 + 3.28i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.296 - 0.913i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 + (0.396 - 1.22i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-12.6 + 9.18i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.44 + 10.6i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.23 - 3.81i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 6.08T + 89T^{2} \)
97 \( 1 + (2.07 + 6.37i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.76256861886789037859617609242, −9.520304070559611680616203990632, −9.036722540507773777923402289800, −8.131679197959229812388168996637, −6.82137196459476222370261764227, −5.56979726808047160057526009673, −4.34568998348293722736503963439, −3.18999331748321746683135069696, −2.04013256358837471058544562983, −0.56318639587929078878575942604, 2.14419102909652436867729686334, 3.89618875782934909559515956033, 5.25721786433785275703426492890, 6.11039126341451156438642466414, 6.87204979872498460374789601254, 7.937849453233553586212674249967, 8.317263029778153119704699767422, 9.584993307048005039926054006444, 10.24388689456939380619254709990, 11.29217832941893039942542093659

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