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Properties

Label 2-495-11.3-c1-0-2
Degree $2$
Conductor $495$
Sign $-0.962 - 0.270i$
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  + (−2.00 + 1.45i)2-s + (1.28 − 3.94i)4-s + (−0.809 − 0.587i)5-s + (−1.35 + 4.17i)7-s + (1.64 + 5.07i)8-s + 2.48·10-s + (3.08 − 1.22i)11-s + (1.72 − 1.25i)13-s + (−3.36 − 10.3i)14-s + (−3.98 − 2.89i)16-s + (3.20 + 2.32i)17-s + (−0.664 − 2.04i)19-s + (−3.35 + 2.43i)20-s + (−4.39 + 6.95i)22-s − 5.16·23-s + ⋯
L(s)  = 1  + (−1.41 + 1.03i)2-s + (0.641 − 1.97i)4-s + (−0.361 − 0.262i)5-s + (−0.513 + 1.57i)7-s + (0.582 + 1.79i)8-s + 0.784·10-s + (0.929 − 0.369i)11-s + (0.479 − 0.348i)13-s + (−0.900 − 2.77i)14-s + (−0.996 − 0.724i)16-s + (0.777 + 0.564i)17-s + (−0.152 − 0.469i)19-s + (−0.750 + 0.545i)20-s + (−0.937 + 1.48i)22-s − 1.07·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.962 - 0.270i$
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: \((2,\ 495,\ (\ :1/2),\ -0.962 - 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0623430 + 0.452945i\)
\(L(\frac12)\) \(\approx\) \(0.0623430 + 0.452945i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.08 + 1.22i)T \)
good2 \( 1 + (2.00 - 1.45i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (1.35 - 4.17i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.72 + 1.25i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.20 - 2.32i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.664 + 2.04i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 5.16T + 23T^{2} \)
29 \( 1 + (2.45 - 7.55i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (5.00 - 3.63i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.64 - 8.15i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.35 - 7.25i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 2.22T + 43T^{2} \)
47 \( 1 + (2.12 + 6.54i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.60 - 3.34i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.16 - 3.59i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.91 + 2.84i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 9.57T + 67T^{2} \)
71 \( 1 + (1.70 + 1.24i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.52 - 13.9i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.63 - 1.91i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-8.62 - 6.26i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 7.64T + 89T^{2} \)
97 \( 1 + (-4.58 + 3.33i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.14366233764325043150295253009, −10.05728341906841530860962811003, −9.229273755884203751877786889797, −8.640762821524387818123447216261, −8.073726235361862274514602594001, −6.80999090395143161622421878980, −6.07449336597262126048637178350, −5.31121240173598387654594412736, −3.37847048359737975806204199152, −1.51530909531972264778653161731, 0.46228619832269552148873043920, 1.86184190349788574265475505569, 3.56167176808713498923545687008, 4.02856901375325205931718585099, 6.28734910302132209290651583934, 7.43146872670595493102554023477, 7.76531645388023524215448845558, 9.109851869847321204463736795023, 9.762309791935824133388276745971, 10.43337020589462589694591809168

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