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Properties

Label 2-495-11.5-c1-0-6
Degree $2$
Conductor $495$
Sign $-0.246 + 0.969i$
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  + (−0.850 − 2.61i)2-s + (−4.51 + 3.27i)4-s + (−0.309 + 0.951i)5-s + (−2.21 + 1.60i)7-s + (7.96 + 5.78i)8-s + 2.75·10-s + (3.02 − 1.36i)11-s + (−0.857 − 2.63i)13-s + (6.08 + 4.41i)14-s + (4.92 − 15.1i)16-s + (1.16 − 3.59i)17-s + (3.43 + 2.49i)19-s + (−1.72 − 5.30i)20-s + (−6.14 − 6.74i)22-s + 1.37·23-s + ⋯
L(s)  = 1  + (−0.601 − 1.85i)2-s + (−2.25 + 1.63i)4-s + (−0.138 + 0.425i)5-s + (−0.835 + 0.606i)7-s + (2.81 + 2.04i)8-s + 0.870·10-s + (0.911 − 0.412i)11-s + (−0.237 − 0.731i)13-s + (1.62 + 1.18i)14-s + (1.23 − 3.78i)16-s + (0.283 − 0.872i)17-s + (0.789 + 0.573i)19-s + (−0.385 − 1.18i)20-s + (−1.31 − 1.43i)22-s + 0.287·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.246 + 0.969i$
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: \((2,\ 495,\ (\ :1/2),\ -0.246 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.507651 - 0.653127i\)
\(L(\frac12)\) \(\approx\) \(0.507651 - 0.653127i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.02 + 1.36i)T \)
good2 \( 1 + (0.850 + 2.61i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + (2.21 - 1.60i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.857 + 2.63i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.16 + 3.59i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.43 - 2.49i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 1.37T + 23T^{2} \)
29 \( 1 + (-7.17 + 5.21i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.68 - 5.19i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.86 + 1.35i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.97 - 5.06i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 12.3T + 43T^{2} \)
47 \( 1 + (4.52 + 3.28i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.168 - 0.517i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.314 + 0.228i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.43 - 13.6i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 3.65T + 67T^{2} \)
71 \( 1 + (-2.83 + 8.71i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (6.60 - 4.79i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.06 + 3.27i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.43 + 4.42i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 6.62T + 89T^{2} \)
97 \( 1 + (5.12 + 15.7i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.67425706655381172378723750101, −9.838180996645959190819328532898, −9.326973059102827012960569740052, −8.409596567964955923014862715892, −7.39637792713415297487248334173, −5.87338142871007958630589107257, −4.43753452143861906330293845800, −3.21534520353465743943001755555, −2.71683384878203476211145047252, −0.919185061644927901798681819061, 0.981216813702362753778636766034, 3.94438154995675476684512413402, 4.76649478105219156183117755962, 5.99918262571794122179208209757, 6.75482898275063956914057366713, 7.40100059100100598133715747164, 8.399252297471502773316175518656, 9.374683213142139642040866695307, 9.663102864065282323878708584847, 10.82860499294962330712748078016

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