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Properties

Label 2-495-11.3-c1-0-12
Degree $2$
Conductor $495$
Sign $0.671 + 0.740i$
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  + (0.918 − 0.667i)2-s + (−0.219 + 0.675i)4-s + (−0.809 − 0.587i)5-s + (1.26 − 3.90i)7-s + (0.951 + 2.92i)8-s − 1.13·10-s + (3.26 + 0.600i)11-s + (1.83 − 1.33i)13-s + (−1.43 − 4.42i)14-s + (1.67 + 1.21i)16-s + (4.06 + 2.95i)17-s + (−2.34 − 7.20i)19-s + (0.574 − 0.417i)20-s + (3.39 − 1.62i)22-s − 1.97·23-s + ⋯
L(s)  = 1  + (0.649 − 0.472i)2-s + (−0.109 + 0.337i)4-s + (−0.361 − 0.262i)5-s + (0.479 − 1.47i)7-s + (0.336 + 1.03i)8-s − 0.359·10-s + (0.983 + 0.180i)11-s + (0.510 − 0.370i)13-s + (−0.384 − 1.18i)14-s + (0.419 + 0.304i)16-s + (0.984 + 0.715i)17-s + (−0.537 − 1.65i)19-s + (0.128 − 0.0933i)20-s + (0.724 − 0.346i)22-s − 0.411·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.671 + 0.740i$
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: \((2,\ 495,\ (\ :1/2),\ 0.671 + 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83741 - 0.813980i\)
\(L(\frac12)\) \(\approx\) \(1.83741 - 0.813980i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.26 - 0.600i)T \)
good2 \( 1 + (-0.918 + 0.667i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (-1.26 + 3.90i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.83 + 1.33i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.06 - 2.95i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.34 + 7.20i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.97T + 23T^{2} \)
29 \( 1 + (-1.23 + 3.80i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.08 + 2.24i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.07 - 6.40i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.24 - 3.82i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 9.57T + 43T^{2} \)
47 \( 1 + (-0.984 - 3.03i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (9.08 - 6.60i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.94 - 9.07i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.92 + 2.12i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 6.31T + 67T^{2} \)
71 \( 1 + (9.94 + 7.22i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.20 - 3.69i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (14.2 - 10.3i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (11.2 + 8.13i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 7.85T + 89T^{2} \)
97 \( 1 + (4.18 - 3.04i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.05286392293724724625507490507, −10.23705593120431753282094128484, −8.952632170184282182475612955944, −8.017831652134352996794987945402, −7.35584157094138405478344553642, −6.09260490297006020809707051996, −4.48503510306844591071952975702, −4.25002380755267902059240173399, −3.07602813881436782163193591716, −1.23864878560500929907227351682, 1.64441345291449648318294270770, 3.40465946397727541295520794226, 4.45499242174187968916871656758, 5.66181551240218284921774394003, 6.08668253832330712078530332871, 7.24437342341580238772183916140, 8.403941396625651768987653676119, 9.185054270327130002775119178064, 10.14789078319272106670457356545, 11.22410597923788342954653238345

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