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Properties

Label 2-495-495.113-c1-0-46
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  + (0.445 + 1.15i)2-s + (0.149 + 1.72i)3-s + (0.339 − 0.305i)4-s + (0.344 − 2.20i)5-s + (−1.93 + 0.941i)6-s + (1.67 − 2.58i)7-s + (2.71 + 1.38i)8-s + (−2.95 + 0.517i)9-s + (2.71 − 0.584i)10-s + (1.09 − 3.13i)11-s + (0.578 + 0.540i)12-s + (1.71 − 2.12i)13-s + (3.74 + 0.795i)14-s + (3.86 + 0.262i)15-s + (−0.300 + 2.86i)16-s + (0.429 + 2.71i)17-s + ⋯
L(s)  = 1  + (0.314 + 0.819i)2-s + (0.0865 + 0.996i)3-s + (0.169 − 0.152i)4-s + (0.153 − 0.988i)5-s + (−0.789 + 0.384i)6-s + (0.634 − 0.976i)7-s + (0.961 + 0.489i)8-s + (−0.985 + 0.172i)9-s + (0.858 − 0.184i)10-s + (0.330 − 0.943i)11-s + (0.167 + 0.155i)12-s + (0.477 − 0.589i)13-s + (1.00 + 0.212i)14-s + (0.997 + 0.0678i)15-s + (−0.0751 + 0.715i)16-s + (0.104 + 0.657i)17-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: $\chi_{495} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.734 - 0.678i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96348 + 0.767926i\)
\(L(\frac12)\) \(\approx\) \(1.96348 + 0.767926i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.149 - 1.72i)T \)
5 \( 1 + (-0.344 + 2.20i)T \)
11 \( 1 + (-1.09 + 3.13i)T \)
good2 \( 1 + (-0.445 - 1.15i)T + (-1.48 + 1.33i)T^{2} \)
7 \( 1 + (-1.67 + 2.58i)T + (-2.84 - 6.39i)T^{2} \)
13 \( 1 + (-1.71 + 2.12i)T + (-2.70 - 12.7i)T^{2} \)
17 \( 1 + (-0.429 - 2.71i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (1.34 - 0.435i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (-1.67 - 6.25i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (6.66 - 1.41i)T + (26.4 - 11.7i)T^{2} \)
31 \( 1 + (0.112 + 1.06i)T + (-30.3 + 6.44i)T^{2} \)
37 \( 1 + (1.68 + 3.31i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (1.30 - 6.11i)T + (-37.4 - 16.6i)T^{2} \)
43 \( 1 + (0.956 - 3.57i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (6.37 - 0.333i)T + (46.7 - 4.91i)T^{2} \)
53 \( 1 + (0.955 - 6.03i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (-9.51 - 10.5i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (0.800 - 7.62i)T + (-59.6 - 12.6i)T^{2} \)
67 \( 1 + (2.62 + 9.78i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.18 - 5.76i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-5.16 + 2.63i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (-1.81 + 4.08i)T + (-52.8 - 58.7i)T^{2} \)
83 \( 1 + (7.00 + 8.64i)T + (-17.2 + 81.1i)T^{2} \)
89 \( 1 - 17.5T + 89T^{2} \)
97 \( 1 + (-2.86 + 1.10i)T + (72.0 - 64.9i)T^{2} \)
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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

−10.96998080427968106748408455173, −10.25750790245237638419666168460, −9.145868810860597967699125658287, −8.257094360031392230881150066025, −7.57635063391742114100005855666, −6.04553222009103674089201814765, −5.47114890636002986622129903824, −4.51249786958859151960728202874, −3.61811443902733014241720195724, −1.38818393514778574225000589281, 1.89121392498791490478729398974, 2.38770975436082980713209176631, 3.62803106885156795024627375215, 5.10346660105958780766388437357, 6.51134102488210187749958960695, 7.02764214848482902324144081266, 8.027245590087560178137936749513, 9.060606317315906421843739959291, 10.22264183613016813978306300811, 11.33667551253713529981883602783

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