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Properties

Label 2-495-15.14-c2-0-14
Degree $2$
Conductor $495$
Sign $-0.0719 - 0.997i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0719 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0719 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.0719 - 0.997i$
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: \((2,\ 495,\ (\ :1),\ -0.0719 - 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.498258582\)
\(L(\frac12)\) \(\approx\) \(2.498258582\)
\(L(2)\) 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

−11.16766087764076070856420947772, −10.03949813003601504213559722789, −9.360007754088120923866783460543, −8.308196385538910488556443255710, −6.86322880596680347788558585969, −6.25712254502170357885337222950, −5.28181992306013118651221159663, −4.25644875839983144609188626900, −3.16903015898810089430677551573, −2.00267892262048990466022686668, 0.71316814823024399461846480126, 2.58857286569899324308711910746, 3.82455830440871400281814164090, 4.80048626839754111044940073752, 5.69348146346772986945239292849, 6.28845660680350658148696125308, 7.938953520483206924390047187152, 8.597714037408933278689520604882, 9.806152493951903943432166272565, 10.35213356769793521712087792762

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