↑

Properties

Label 2-495-55.28-c1-0-7
Degree $2$
Conductor $495$
Sign $0.735 - 0.677i$
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  + (0.188 − 0.369i)2-s + (1.07 + 1.47i)4-s + (−2.23 − 0.00601i)5-s + (4.21 + 0.667i)7-s + (1.56 − 0.248i)8-s + (−0.423 + 0.825i)10-s + (−0.957 + 3.17i)11-s + (−1.20 − 0.612i)13-s + (1.04 − 1.43i)14-s + (−0.926 + 2.85i)16-s + (−1.39 + 0.708i)17-s + (−3.55 − 2.58i)19-s + (−2.39 − 3.31i)20-s + (0.993 + 0.951i)22-s + (6.71 + 6.71i)23-s + ⋯
L(s)  = 1  + (0.133 − 0.261i)2-s + (0.537 + 0.739i)4-s + (−0.999 − 0.00269i)5-s + (1.59 + 0.252i)7-s + (0.554 − 0.0877i)8-s + (−0.133 + 0.260i)10-s + (−0.288 + 0.957i)11-s + (−0.333 − 0.169i)13-s + (0.278 − 0.382i)14-s + (−0.231 + 0.712i)16-s + (−0.337 + 0.171i)17-s + (−0.815 − 0.592i)19-s + (−0.535 − 0.740i)20-s + (0.211 + 0.202i)22-s + (1.40 + 1.40i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.735 - 0.677i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ 0.735 - 0.677i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55521 + 0.606557i\)
\(L(\frac12)\) \(\approx\) \(1.55521 + 0.606557i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.00601i)T \)
11 \( 1 + (0.957 - 3.17i)T \)
good2 \( 1 + (-0.188 + 0.369i)T + (-1.17 - 1.61i)T^{2} \)
7 \( 1 + (-4.21 - 0.667i)T + (6.65 + 2.16i)T^{2} \)
13 \( 1 + (1.20 + 0.612i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (1.39 - 0.708i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (3.55 + 2.58i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-6.71 - 6.71i)T + 23iT^{2} \)
29 \( 1 + (-5.75 + 4.17i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.83 - 5.64i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.411 + 2.59i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (-3.97 + 5.47i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (2.05 - 2.05i)T - 43iT^{2} \)
47 \( 1 + (6.52 - 1.03i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (-0.813 + 1.59i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (5.40 + 7.43i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (9.79 + 3.18i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (4.87 - 4.87i)T - 67iT^{2} \)
71 \( 1 + (-3.98 + 12.2i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.45 + 9.18i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (2.52 + 7.77i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.08 - 8.01i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 + 0.813iT - 89T^{2} \)
97 \( 1 + (9.74 + 4.96i)T + (57.0 + 78.4i)T^{2} \)
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

−11.12272994047261026780583925336, −10.65328383871533666634895907178, −9.051653843756032071716594200034, −8.112360820881143713501158690896, −7.60504444312216256273226743799, −6.77655427722969948167029105338, −4.92678241679129684173282800738, −4.43751437965028067885004712321, −3.04264694606700264040795899607, −1.80701172079899105640736888506, 1.07636976188576078404204046207, 2.68233767552236309262211594539, 4.43683364076584700440166473294, 4.96216034836698904666909394076, 6.26921139583030373500914850440, 7.22189367872478239618970926787, 8.114458253421773342327902257287, 8.709202090197423953777135207893, 10.32800972916048790642445601033, 11.01865723967662664900441489272

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