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Properties

Label 2-495-45.32-c1-0-7
Degree $2$
Conductor $495$
Sign $0.533 - 0.845i$
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  + (−2.37 − 0.635i)2-s + (−1.63 − 0.561i)3-s + (3.49 + 2.01i)4-s + (−0.239 + 2.22i)5-s + (3.53 + 2.37i)6-s + (0.646 − 2.41i)7-s + (−3.53 − 3.53i)8-s + (2.37 + 1.83i)9-s + (1.98 − 5.12i)10-s + (−0.866 + 0.5i)11-s + (−4.59 − 5.26i)12-s + (0.341 + 1.27i)13-s + (−3.06 + 5.31i)14-s + (1.64 − 3.50i)15-s + (2.10 + 3.64i)16-s + (2.19 − 2.19i)17-s + ⋯
L(s)  = 1  + (−1.67 − 0.449i)2-s + (−0.946 − 0.323i)3-s + (1.74 + 1.00i)4-s + (−0.107 + 0.994i)5-s + (1.44 + 0.968i)6-s + (0.244 − 0.911i)7-s + (−1.24 − 1.24i)8-s + (0.790 + 0.612i)9-s + (0.626 − 1.62i)10-s + (−0.261 + 0.150i)11-s + (−1.32 − 1.52i)12-s + (0.0946 + 0.353i)13-s + (−0.819 + 1.42i)14-s + (0.423 − 0.905i)15-s + (0.526 + 0.911i)16-s + (0.531 − 0.531i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.533 - 0.845i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.533 - 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.287626 + 0.158641i\)
\(L(\frac12)\) \(\approx\) \(0.287626 + 0.158641i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.63 + 0.561i)T \)
5 \( 1 + (0.239 - 2.22i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
good2 \( 1 + (2.37 + 0.635i)T + (1.73 + i)T^{2} \)
7 \( 1 + (-0.646 + 2.41i)T + (-6.06 - 3.5i)T^{2} \)
13 \( 1 + (-0.341 - 1.27i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-2.19 + 2.19i)T - 17iT^{2} \)
19 \( 1 + 1.23iT - 19T^{2} \)
23 \( 1 + (8.47 - 2.27i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.63 - 2.83i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.649 - 1.12i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.35 - 6.35i)T + 37iT^{2} \)
41 \( 1 + (-1.08 - 0.626i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.62 - 0.971i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-5.05 - 1.35i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.85 - 2.85i)T + 53iT^{2} \)
59 \( 1 + (-2.30 + 3.99i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.74 - 11.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.83 - 1.83i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.83iT - 71T^{2} \)
73 \( 1 + (7.08 - 7.08i)T - 73iT^{2} \)
79 \( 1 + (14.7 - 8.54i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.77 - 10.3i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 13.9T + 89T^{2} \)
97 \( 1 + (3.81 - 14.2i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.95346390024864254923643350470, −10.16347006842527447126845749539, −9.790937866583839593899849348696, −8.267905171867415605833210942331, −7.39798031036228562661807756256, −7.04351259875676532519389900559, −5.89005948885818153224357782132, −4.17775120215623550182507325076, −2.55695192159532491592966131985, −1.17534726334793954705759357273, 0.44516734403023178333448970139, 1.90931158456581320286596828689, 4.25353105249752253816733699971, 5.76639454533798727629009554689, 5.93720747032240216296537456876, 7.50325493328734427214790750165, 8.251650632364683477997872668098, 8.983947514927894498560282846500, 9.866605875681421182834572993483, 10.45974292663908198753874924186

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