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Properties

Label 2-495-45.2-c1-0-34
Degree $2$
Conductor $495$
Sign $0.961 - 0.273i$
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  + (0.260 + 0.971i)2-s + (1.09 − 1.34i)3-s + (0.855 − 0.493i)4-s + (−1.65 + 1.50i)5-s + (1.59 + 0.709i)6-s + (0.704 − 0.188i)7-s + (2.12 + 2.12i)8-s + (−0.619 − 2.93i)9-s + (−1.89 − 1.21i)10-s + (0.866 + 0.5i)11-s + (0.268 − 1.68i)12-s + (4.50 + 1.20i)13-s + (0.367 + 0.635i)14-s + (0.216 + 3.86i)15-s + (−0.524 + 0.908i)16-s + (0.213 − 0.213i)17-s + ⋯
L(s)  = 1  + (0.184 + 0.687i)2-s + (0.629 − 0.776i)3-s + (0.427 − 0.246i)4-s + (−0.740 + 0.672i)5-s + (0.649 + 0.289i)6-s + (0.266 − 0.0713i)7-s + (0.751 + 0.751i)8-s + (−0.206 − 0.978i)9-s + (−0.598 − 0.384i)10-s + (0.261 + 0.150i)11-s + (0.0775 − 0.487i)12-s + (1.24 + 0.334i)13-s + (0.0981 + 0.169i)14-s + (0.0559 + 0.998i)15-s + (−0.131 + 0.227i)16-s + (0.0518 − 0.0518i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 - 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.961 - 0.273i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (452, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.961 - 0.273i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11742 + 0.295511i\)
\(L(\frac12)\) \(\approx\) \(2.11742 + 0.295511i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.09 + 1.34i)T \)
5 \( 1 + (1.65 - 1.50i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
good2 \( 1 + (-0.260 - 0.971i)T + (-1.73 + i)T^{2} \)
7 \( 1 + (-0.704 + 0.188i)T + (6.06 - 3.5i)T^{2} \)
13 \( 1 + (-4.50 - 1.20i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-0.213 + 0.213i)T - 17iT^{2} \)
19 \( 1 + 2.46iT - 19T^{2} \)
23 \( 1 + (0.0257 - 0.0961i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.87 - 3.24i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.41 - 2.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.548 + 0.548i)T + 37iT^{2} \)
41 \( 1 + (1.82 - 1.05i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.97 + 7.37i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.25 + 4.68i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (5.68 + 5.68i)T + 53iT^{2} \)
59 \( 1 + (-2.72 - 4.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.748 - 1.29i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.55 - 13.2i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 13.6iT - 71T^{2} \)
73 \( 1 + (1.56 - 1.56i)T - 73iT^{2} \)
79 \( 1 + (5.77 + 3.33i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.43 - 1.72i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 + (9.74 - 2.61i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.26528903267669234558020045004, −10.18503574395822333500800921891, −8.737169646466228137394471839312, −8.140387168253893427197419286520, −7.03902812490947921092063588751, −6.80872704883862108580945429705, −5.68135616444315004702839418258, −4.14873479224992889612104907446, −2.95983924542802521119362737751, −1.55570472705942870937383991314, 1.58016072311596226372994777256, 3.16100818011969028222847870804, 3.86886171046281169142024586831, 4.75440245722376160011885298805, 6.16342308357465534194433518894, 7.73130248042702167948859579314, 8.185324665896910438027821153520, 9.167026390321682723620297920783, 10.15105555180181683613472376205, 11.12403029363550611861737222775

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