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Properties

Label 2-495-11.4-c1-0-15
Degree $2$
Conductor $495$
Sign $0.717 + 0.696i$
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  + (1.09 + 0.796i)2-s + (−0.0501 − 0.154i)4-s + (−0.809 + 0.587i)5-s + (−1.12 − 3.47i)7-s + (0.905 − 2.78i)8-s − 1.35·10-s + (−0.490 − 3.28i)11-s + (2.29 + 1.66i)13-s + (1.52 − 4.70i)14-s + (2.95 − 2.14i)16-s + (2.98 − 2.17i)17-s + (−0.0293 + 0.0904i)19-s + (0.131 + 0.0953i)20-s + (2.07 − 3.98i)22-s − 1.16·23-s + ⋯
L(s)  = 1  + (0.775 + 0.563i)2-s + (−0.0250 − 0.0771i)4-s + (−0.361 + 0.262i)5-s + (−0.426 − 1.31i)7-s + (0.320 − 0.985i)8-s − 0.428·10-s + (−0.147 − 0.989i)11-s + (0.635 + 0.461i)13-s + (0.408 − 1.25i)14-s + (0.738 − 0.536i)16-s + (0.724 − 0.526i)17-s + (−0.00674 + 0.0207i)19-s + (0.0293 + 0.0213i)20-s + (0.442 − 0.850i)22-s − 0.242·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.717 + 0.696i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.717 + 0.696i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65570 - 0.671245i\)
\(L(\frac12)\) \(\approx\) \(1.65570 - 0.671245i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (0.490 + 3.28i)T \)
good2 \( 1 + (-1.09 - 0.796i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 + (1.12 + 3.47i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-2.29 - 1.66i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.98 + 2.17i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.0293 - 0.0904i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 1.16T + 23T^{2} \)
29 \( 1 + (-2.08 - 6.42i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (5.48 + 3.98i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.04 - 9.35i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.57 + 7.91i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.96T + 43T^{2} \)
47 \( 1 + (-0.687 + 2.11i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.42 - 1.75i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.62 - 8.09i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-6.86 + 4.98i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 13.4T + 67T^{2} \)
71 \( 1 + (-6.71 + 4.88i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.407 + 1.25i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-11.2 - 8.15i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (8.61 - 6.25i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + (-3.50 - 2.54i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.74105118790806919293268331342, −10.16158107367248111637709134222, −9.053929858815882764768654597867, −7.78925872247634321956513076737, −6.98670374917868694638686934266, −6.23631861615225059803040337740, −5.19460986192352640657110055662, −4.00194089725673645162383684774, −3.37588312822167555822831899414, −0.897360236173824153839705408948, 2.06103613319295832276918751751, 3.16633436787110400176932222519, 4.18788731185390127366834752935, 5.27961314380454245565569368119, 6.06451384921922744773464451878, 7.59283270016965545150509426104, 8.372646654736167433170729457286, 9.263442257355262086715645489403, 10.31702708462147198260860707695, 11.40184914787243853789531997611

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