↑

Properties

Label 2-495-55.7-c1-0-23
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: $\chi_{495} (172, \cdot )$
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

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

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