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Properties

Label 2-495-11.3-c1-0-7
Degree $2$
Conductor $495$
Sign $0.734 - 0.678i$
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  + (1.08 − 0.786i)2-s + (−0.0646 + 0.198i)4-s + (0.809 + 0.587i)5-s + (−1.16 + 3.59i)7-s + (0.913 + 2.81i)8-s + 1.33·10-s + (0.569 − 3.26i)11-s + (−3.99 + 2.90i)13-s + (1.56 + 4.81i)14-s + (2.86 + 2.07i)16-s + (2.63 + 1.91i)17-s + (0.424 + 1.30i)19-s + (−0.169 + 0.122i)20-s + (−1.95 − 3.98i)22-s + 1.93·23-s + ⋯
L(s)  = 1  + (0.765 − 0.556i)2-s + (−0.0323 + 0.0994i)4-s + (0.361 + 0.262i)5-s + (−0.441 + 1.35i)7-s + (0.322 + 0.994i)8-s + 0.423·10-s + (0.171 − 0.985i)11-s + (−1.10 + 0.804i)13-s + (0.418 + 1.28i)14-s + (0.715 + 0.519i)16-s + (0.639 + 0.464i)17-s + (0.0973 + 0.299i)19-s + (−0.0378 + 0.0274i)20-s + (−0.416 − 0.849i)22-s + 0.403·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.734 - 0.678i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.734 - 0.678i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84749 + 0.723213i\)
\(L(\frac12)\) \(\approx\) \(1.84749 + 0.723213i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.569 + 3.26i)T \)
good2 \( 1 + (-1.08 + 0.786i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (1.16 - 3.59i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (3.99 - 2.90i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.63 - 1.91i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.424 - 1.30i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 1.93T + 23T^{2} \)
29 \( 1 + (-0.537 + 1.65i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.79 + 3.48i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.45 + 4.46i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.91 + 8.97i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 + (0.248 + 0.766i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-8.65 + 6.28i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.75 - 11.5i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (6.95 + 5.04i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 2.77T + 67T^{2} \)
71 \( 1 + (2.21 + 1.61i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.89 - 8.91i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (10.3 - 7.50i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.76 + 2.00i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 7.85T + 89T^{2} \)
97 \( 1 + (-11.0 + 8.00i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.37850144415600103510994481278, −10.29398584034989621617046394003, −9.239422810026256145125492569254, −8.562581265234193821579956615823, −7.39159834354102616837707625995, −6.01726367387223161249220909164, −5.45595848292581388041468684693, −4.15840056705695328135033777950, −2.98266556787675717500117576212, −2.21078882039751293094064844941, 1.00302664681717947949726902982, 3.07823418354416514155313359272, 4.46433867675487715577867770297, 4.97024871660925096860831724611, 6.18441970865218455534692735674, 7.13604101188695228750790334358, 7.62864295005505872358571560694, 9.331381673278104265624462127701, 10.05251095888336884321803090711, 10.48366744699893804757737342794

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