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Properties

Label 2-495-15.14-c2-0-3
Degree $2$
Conductor $495$
Sign $-0.283 - 0.958i$
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  − 0.284·2-s − 3.91·4-s + (−3.92 − 3.09i)5-s − 5.75i·7-s + 2.25·8-s + (1.11 + 0.879i)10-s − 3.31i·11-s − 4.74i·13-s + 1.63i·14-s + 15.0·16-s − 8.57·17-s − 2.79·19-s + (15.3 + 12.1i)20-s + 0.942i·22-s − 29.9·23-s + ⋯
L(s)  = 1  − 0.142·2-s − 0.979·4-s + (−0.785 − 0.619i)5-s − 0.822i·7-s + 0.281·8-s + (0.111 + 0.0879i)10-s − 0.301i·11-s − 0.365i·13-s + 0.116i·14-s + 0.939·16-s − 0.504·17-s − 0.147·19-s + (0.769 + 0.606i)20-s + 0.0428i·22-s − 1.30·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.283 - 0.958i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.283 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.283 - 0.958i$
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: \((2,\ 495,\ (\ :1),\ -0.283 - 0.958i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2124650211\)
\(L(\frac12)\) \(\approx\) \(0.2124650211\)
\(L(2)\) 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 \( 1 + (3.92 + 3.09i)T \)
11 \( 1 + 3.31iT \)
good2 \( 1 + 0.284T + 4T^{2} \)
7 \( 1 + 5.75iT - 49T^{2} \)
13 \( 1 + 4.74iT - 169T^{2} \)
17 \( 1 + 8.57T + 289T^{2} \)
19 \( 1 + 2.79T + 361T^{2} \)
23 \( 1 + 29.9T + 529T^{2} \)
29 \( 1 - 33.8iT - 841T^{2} \)
31 \( 1 - 28.7T + 961T^{2} \)
37 \( 1 - 11.3iT - 1.36e3T^{2} \)
41 \( 1 - 15.2iT - 1.68e3T^{2} \)
43 \( 1 - 76.6iT - 1.84e3T^{2} \)
47 \( 1 + 57.5T + 2.20e3T^{2} \)
53 \( 1 + 72.2T + 2.80e3T^{2} \)
59 \( 1 - 52.4iT - 3.48e3T^{2} \)
61 \( 1 - 101.T + 3.72e3T^{2} \)
67 \( 1 + 33.5iT - 4.48e3T^{2} \)
71 \( 1 + 38.3iT - 5.04e3T^{2} \)
73 \( 1 + 101. iT - 5.32e3T^{2} \)
79 \( 1 + 66.6T + 6.24e3T^{2} \)
83 \( 1 - 106.T + 6.88e3T^{2} \)
89 \( 1 - 98.0iT - 7.92e3T^{2} \)
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

−10.91155642897925519897830095585, −10.04927623986739030922422714122, −9.153957704997353329544584383847, −8.210638920774382594575574644917, −7.76481755591406101763485721118, −6.41287894892719929701601662223, −5.03684279986116522546681132379, −4.32018140026341845930954786818, −3.39585528567487514265514417263, −1.12471738948794690680155489247, 0.10953246495900636849726088353, 2.27831893329113725773666724801, 3.73957798758267717090472632366, 4.54690012258259150413669470344, 5.76335407128824257245491219640, 6.84668300639804071809118489654, 8.018683760109436551920295857074, 8.557054738911884828700148960852, 9.605100535429437801670049136333, 10.31675783288706416718904576971

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