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Properties

Label 2-495-11.4-c1-0-16
Degree $2$
Conductor $495$
Sign $-0.690 + 0.723i$
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  + (−0.392 − 0.285i)2-s + (−0.545 − 1.67i)4-s + (0.809 − 0.587i)5-s + (−0.500 − 1.54i)7-s + (−0.564 + 1.73i)8-s − 0.485·10-s + (3.27 − 0.518i)11-s + (−4.01 − 2.91i)13-s + (−0.242 + 0.747i)14-s + (−2.13 + 1.55i)16-s + (2.53 − 1.84i)17-s + (0.339 − 1.04i)19-s + (−1.42 − 1.03i)20-s + (−1.43 − 0.730i)22-s − 6.75·23-s + ⋯
L(s)  = 1  + (−0.277 − 0.201i)2-s + (−0.272 − 0.839i)4-s + (0.361 − 0.262i)5-s + (−0.189 − 0.582i)7-s + (−0.199 + 0.614i)8-s − 0.153·10-s + (0.987 − 0.156i)11-s + (−1.11 − 0.808i)13-s + (−0.0648 + 0.199i)14-s + (−0.534 + 0.388i)16-s + (0.615 − 0.447i)17-s + (0.0778 − 0.239i)19-s + (−0.319 − 0.231i)20-s + (−0.305 − 0.155i)22-s − 1.40·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.690 + 0.723i$
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: \((2,\ 495,\ (\ :1/2),\ -0.690 + 0.723i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.380884 - 0.889652i\)
\(L(\frac12)\) \(\approx\) \(0.380884 - 0.889652i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.27 + 0.518i)T \)
good2 \( 1 + (0.392 + 0.285i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 + (0.500 + 1.54i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (4.01 + 2.91i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.53 + 1.84i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.339 + 1.04i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 6.75T + 23T^{2} \)
29 \( 1 + (-1.09 - 3.37i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.90 + 5.01i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.65 + 5.09i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.64 - 5.05i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.41T + 43T^{2} \)
47 \( 1 + (-2.03 + 6.27i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (7.86 + 5.71i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.22 - 6.83i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-5.36 + 3.89i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 2.36T + 67T^{2} \)
71 \( 1 + (-13.4 + 9.79i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.62 - 8.07i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.61 - 1.17i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (7.72 - 5.61i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 2.65T + 89T^{2} \)
97 \( 1 + (-11.7 - 8.53i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.37278578331987936339355696102, −9.768401501566243894467751135908, −9.154343809181142635271266034985, −7.989038608305003087255093980275, −6.92211938820793729214988815344, −5.81786848601454927385021986056, −5.03341998001700814987898899864, −3.77448644757659633451344285379, −2.10981022581869792350408146394, −0.63642707231976803205375552571, 2.06948146311594633960458441549, 3.44020409589580738472071921137, 4.46820995021713815054847261816, 5.86678642280845221180718564256, 6.81542479824840282476342565970, 7.64822095058357106624823568143, 8.682857218145420136375024651606, 9.445066228297916981355872510221, 10.05372676525801938569890726551, 11.47475012366628712297591317173

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