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Properties

Label 2-495-45.4-c1-0-37
Degree $2$
Conductor $495$
Sign $0.840 + 0.541i$
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  + (−1.32 + 0.763i)2-s + (−0.144 − 1.72i)3-s + (0.164 − 0.285i)4-s + (2.06 + 0.851i)5-s + (1.50 + 2.17i)6-s + (3.72 − 2.15i)7-s − 2.54i·8-s + (−2.95 + 0.498i)9-s + (−3.38 + 0.452i)10-s + (−0.5 − 0.866i)11-s + (−0.516 − 0.243i)12-s + (2.08 + 1.20i)13-s + (−3.28 + 5.68i)14-s + (1.17 − 3.69i)15-s + (2.27 + 3.94i)16-s − 0.975i·17-s + ⋯
L(s)  = 1  + (−0.934 + 0.539i)2-s + (−0.0833 − 0.996i)3-s + (0.0823 − 0.142i)4-s + (0.924 + 0.380i)5-s + (0.615 + 0.886i)6-s + (1.40 − 0.813i)7-s − 0.901i·8-s + (−0.986 + 0.166i)9-s + (−1.06 + 0.143i)10-s + (−0.150 − 0.261i)11-s + (−0.149 − 0.0702i)12-s + (0.578 + 0.334i)13-s + (−0.877 + 1.52i)14-s + (0.302 − 0.953i)15-s + (0.568 + 0.985i)16-s − 0.236i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.840 + 0.541i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.840 + 0.541i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00787 - 0.296787i\)
\(L(\frac12)\) \(\approx\) \(1.00787 - 0.296787i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.144 + 1.72i)T \)
5 \( 1 + (-2.06 - 0.851i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (1.32 - 0.763i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-3.72 + 2.15i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (-2.08 - 1.20i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.975iT - 17T^{2} \)
19 \( 1 + 1.61T + 19T^{2} \)
23 \( 1 + (5.95 + 3.43i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.29 - 3.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.81 + 8.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.88iT - 37T^{2} \)
41 \( 1 + (2.71 - 4.69i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.45 + 3.14i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.78 + 2.76i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.92iT - 53T^{2} \)
59 \( 1 + (-1.92 + 3.32i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.71 - 8.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.0 - 5.79i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.77T + 71T^{2} \)
73 \( 1 - 3.30iT - 73T^{2} \)
79 \( 1 + (1.22 + 2.12i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.4 - 6.59i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.13T + 89T^{2} \)
97 \( 1 + (7.15 - 4.13i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.76583809251986244437841252973, −9.927302869001815862733329513616, −8.687129708760897927656020244882, −8.177674810200628016027241678802, −7.31646880007542805724477992302, −6.57754952277673782236079877092, −5.63576399965432510983548859059, −4.11817269366866030162339800091, −2.24131295932357873363347397431, −1.00084734376130970398895126762, 1.50703378142269086320306774136, 2.60063723995105879071188899471, 4.50424986169403308160365299703, 5.32465876195781472725200724218, 6.00972612240658223787482366903, 8.183003206672621579215270470783, 8.490085036354192403715258172897, 9.362828633426822968079522261548, 10.14274770908200374178727210401, 10.72373095150324257813922292925

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