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Properties

Label 2-495-15.14-c2-0-11
Degree $2$
Conductor $495$
Sign $0.964 - 0.264i$
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  − 2.09·2-s + 0.373·4-s + (−3.17 + 3.86i)5-s − 2.11i·7-s + 7.58·8-s + (6.63 − 8.08i)10-s + 3.31i·11-s − 17.4i·13-s + 4.42i·14-s − 17.3·16-s − 17.3·17-s − 16.6·19-s + (−1.18 + 1.44i)20-s − 6.93i·22-s + 17.6·23-s + ⋯
L(s)  = 1  − 1.04·2-s + 0.0934·4-s + (−0.634 + 0.772i)5-s − 0.302i·7-s + 0.947·8-s + (0.663 − 0.808i)10-s + 0.301i·11-s − 1.34i·13-s + 0.316i·14-s − 1.08·16-s − 1.02·17-s − 0.878·19-s + (−0.0593 + 0.0722i)20-s − 0.315i·22-s + 0.766·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.264i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.964 - 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.964 - 0.264i$
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: \((2,\ 495,\ (\ :1),\ 0.964 - 0.264i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6593546642\)
\(L(\frac12)\) \(\approx\) \(0.6593546642\)
\(L(2)\) 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 \( 1 + (3.17 - 3.86i)T \)
11 \( 1 - 3.31iT \)
good2 \( 1 + 2.09T + 4T^{2} \)
7 \( 1 + 2.11iT - 49T^{2} \)
13 \( 1 + 17.4iT - 169T^{2} \)
17 \( 1 + 17.3T + 289T^{2} \)
19 \( 1 + 16.6T + 361T^{2} \)
23 \( 1 - 17.6T + 529T^{2} \)
29 \( 1 - 38.5iT - 841T^{2} \)
31 \( 1 - 49.2T + 961T^{2} \)
37 \( 1 - 12.4iT - 1.36e3T^{2} \)
41 \( 1 + 1.84iT - 1.68e3T^{2} \)
43 \( 1 + 19.5iT - 1.84e3T^{2} \)
47 \( 1 - 59.7T + 2.20e3T^{2} \)
53 \( 1 + 5.90T + 2.80e3T^{2} \)
59 \( 1 + 80.0iT - 3.48e3T^{2} \)
61 \( 1 - 52.2T + 3.72e3T^{2} \)
67 \( 1 - 74.7iT - 4.48e3T^{2} \)
71 \( 1 - 75.9iT - 5.04e3T^{2} \)
73 \( 1 - 9.20iT - 5.32e3T^{2} \)
79 \( 1 - 123.T + 6.24e3T^{2} \)
83 \( 1 - 65.5T + 6.88e3T^{2} \)
89 \( 1 - 94.2iT - 7.92e3T^{2} \)
97 \( 1 - 92.6iT - 9.40e3T^{2} \)
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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

−10.60367619426331844384866576871, −10.05993074092352940243734942832, −8.834148426277455740294958078471, −8.198026405980120398433351822795, −7.30809496576369125556789707986, −6.58857787486670922420193869647, −4.98292573557621814504657723135, −3.90626338632029340802782170201, −2.53323959611944311841795684803, −0.70434525409978444754773609098, 0.67128750089336567366670673754, 2.14591000834199793546957527412, 4.14646632994943619581252202730, 4.71828680350169557783292224085, 6.28086997019672117212004456307, 7.33713782537167280067785065938, 8.327453626761267769424873352763, 8.890567093100739553690466807409, 9.463131166506933199220684226394, 10.64094364331327070405118567389

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