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Properties

Label 2-495-11.3-c1-0-11
Degree $2$
Conductor $495$
Sign $0.993 - 0.110i$
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  + (0.0756 − 0.0549i)2-s + (−0.615 + 1.89i)4-s + (0.809 + 0.587i)5-s + (1.39 − 4.30i)7-s + (0.115 + 0.354i)8-s + 0.0935·10-s + (2.39 − 2.29i)11-s + (0.924 − 0.671i)13-s + (−0.130 − 0.402i)14-s + (−3.19 − 2.32i)16-s + (2.72 + 1.98i)17-s + (1.88 + 5.78i)19-s + (−1.61 + 1.17i)20-s + (0.0554 − 0.305i)22-s + 5.45·23-s + ⋯
L(s)  = 1  + (0.0534 − 0.0388i)2-s + (−0.307 + 0.946i)4-s + (0.361 + 0.262i)5-s + (0.528 − 1.62i)7-s + (0.0407 + 0.125i)8-s + 0.0295·10-s + (0.723 − 0.690i)11-s + (0.256 − 0.186i)13-s + (−0.0349 − 0.107i)14-s + (−0.798 − 0.580i)16-s + (0.661 + 0.480i)17-s + (0.431 + 1.32i)19-s + (−0.360 + 0.261i)20-s + (0.0118 − 0.0650i)22-s + 1.13·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.993 - 0.110i$
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: \((2,\ 495,\ (\ :1/2),\ 0.993 - 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61554 + 0.0893222i\)
\(L(\frac12)\) \(\approx\) \(1.61554 + 0.0893222i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.39 + 2.29i)T \)
good2 \( 1 + (-0.0756 + 0.0549i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + (-1.39 + 4.30i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.924 + 0.671i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.72 - 1.98i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.88 - 5.78i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 5.45T + 23T^{2} \)
29 \( 1 + (1.02 - 3.15i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.44 - 1.05i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.460 + 1.41i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.539 - 1.66i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 0.263T + 43T^{2} \)
47 \( 1 + (2.13 + 6.58i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.16 - 0.846i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.18 + 6.72i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.02 + 1.47i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 0.516T + 67T^{2} \)
71 \( 1 + (8.68 + 6.30i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.75 - 5.40i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-9.14 + 6.64i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.62 - 2.63i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + (-2.71 + 1.97i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.93735787192692673789091173288, −10.21802556053850054667594831116, −9.117178472249356574909105928963, −8.114484557420123427867405838323, −7.46210475680539746130346061440, −6.52216262762359809653325351148, −5.16853364456865502124298993205, −3.88909692121559615266172526868, −3.36872545623079592588262841395, −1.29692810728268680354357732756, 1.39597753843246940710286979039, 2.63133917044407425843298711651, 4.56916356801682272099551608744, 5.28244167626468068337812896343, 6.05688181102909817846604664672, 7.13823053192590565593016572813, 8.603885290391999653339931509151, 9.253830683444734760219460575824, 9.711131258166031855345551084544, 11.06822446216870866219618467250

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