↑

Properties

Label 2-495-55.8-c1-0-16
Degree $2$
Conductor $495$
Sign $0.779 - 0.626i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.40 + 0.380i)2-s + (3.72 + 1.21i)4-s + (−1.85 + 1.25i)5-s + (2.30 − 1.17i)7-s + (4.15 + 2.11i)8-s + (−4.92 + 2.31i)10-s + (2.55 + 2.11i)11-s + (−0.439 + 2.77i)13-s + (5.97 − 1.94i)14-s + (2.84 + 2.06i)16-s + (−0.147 − 0.932i)17-s + (−1.28 − 3.94i)19-s + (−8.41 + 2.43i)20-s + (5.33 + 6.05i)22-s + (−0.104 − 0.104i)23-s + ⋯
L(s)  = 1  + (1.69 + 0.269i)2-s + (1.86 + 0.605i)4-s + (−0.827 + 0.561i)5-s + (0.869 − 0.443i)7-s + (1.46 + 0.748i)8-s + (−1.55 + 0.731i)10-s + (0.770 + 0.637i)11-s + (−0.121 + 0.769i)13-s + (1.59 − 0.518i)14-s + (0.710 + 0.516i)16-s + (−0.0358 − 0.226i)17-s + (−0.294 − 0.905i)19-s + (−1.88 + 0.544i)20-s + (1.13 + 1.29i)22-s + (−0.0218 − 0.0218i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.779 - 0.626i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.779 - 0.626i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.27664 + 1.15355i\)
\(L(\frac12)\) \(\approx\) \(3.27664 + 1.15355i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.85 - 1.25i)T \)
11 \( 1 + (-2.55 - 2.11i)T \)
good2 \( 1 + (-2.40 - 0.380i)T + (1.90 + 0.618i)T^{2} \)
7 \( 1 + (-2.30 + 1.17i)T + (4.11 - 5.66i)T^{2} \)
13 \( 1 + (0.439 - 2.77i)T + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (0.147 + 0.932i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (1.28 + 3.94i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.104 + 0.104i)T + 23iT^{2} \)
29 \( 1 + (-2.14 + 6.60i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (7.33 - 5.32i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.44 + 6.77i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (3.27 - 1.06i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (3.91 - 3.91i)T - 43iT^{2} \)
47 \( 1 + (-0.942 - 0.479i)T + (27.6 + 38.0i)T^{2} \)
53 \( 1 + (-4.07 - 0.644i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (6.16 + 2.00i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-5.59 + 7.70i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.94 + 2.94i)T - 67iT^{2} \)
71 \( 1 + (-1.02 - 0.744i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.804 - 1.57i)T + (-42.9 + 59.0i)T^{2} \)
79 \( 1 + (-3.51 + 2.55i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-8.18 + 1.29i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 - 4.23iT - 89T^{2} \)
97 \( 1 + (2.25 - 14.2i)T + (-92.2 - 29.9i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.40372187661305116764888589809, −10.70027832133098997284489795374, −9.209946252616271710323274856290, −7.86996815563422843626497860229, −7.03125347134616389219253944097, −6.51595767227446490880237879964, −5.02339114992143349604967768076, −4.35053948542519649324702758424, −3.57638754329494090316644135963, −2.15250601841020089204261437409, 1.65931326534175385175538447303, 3.28199444744181103008801093293, 4.04966305149614283517282266775, 5.09485666091066189901284104869, 5.71406534657435902193127885002, 6.92761450847166400060502071205, 8.107656286003249163649740235024, 8.838815455934612628114142560431, 10.46336966301261183585247850111, 11.31259786317356764406657902784

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