↑

Properties

Label 2-495-45.32-c1-0-49
Degree $2$
Conductor $495$
Sign $0.762 + 0.646i$
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  + (0.901 + 0.241i)2-s + (0.720 + 1.57i)3-s + (−0.977 − 0.564i)4-s + (0.431 − 2.19i)5-s + (0.268 + 1.59i)6-s + (1.07 − 3.99i)7-s + (−2.06 − 2.06i)8-s + (−1.96 + 2.26i)9-s + (0.919 − 1.87i)10-s + (0.866 − 0.5i)11-s + (0.184 − 1.94i)12-s + (−0.0948 − 0.354i)13-s + (1.92 − 3.34i)14-s + (3.76 − 0.899i)15-s + (−0.233 − 0.405i)16-s + (4.85 − 4.85i)17-s + ⋯
L(s)  = 1  + (0.637 + 0.170i)2-s + (0.415 + 0.909i)3-s + (−0.488 − 0.282i)4-s + (0.193 − 0.981i)5-s + (0.109 + 0.650i)6-s + (0.404 − 1.50i)7-s + (−0.730 − 0.730i)8-s + (−0.654 + 0.756i)9-s + (0.290 − 0.592i)10-s + (0.261 − 0.150i)11-s + (0.0533 − 0.561i)12-s + (−0.0263 − 0.0981i)13-s + (0.515 − 0.893i)14-s + (0.972 − 0.232i)15-s + (−0.0584 − 0.101i)16-s + (1.17 − 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $0.762 + 0.646i$
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: \((2,\ 495,\ (\ :1/2),\ 0.762 + 0.646i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81489 - 0.665472i\)
\(L(\frac12)\) \(\approx\) \(1.81489 - 0.665472i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.720 - 1.57i)T \)
5 \( 1 + (-0.431 + 2.19i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
good2 \( 1 + (-0.901 - 0.241i)T + (1.73 + i)T^{2} \)
7 \( 1 + (-1.07 + 3.99i)T + (-6.06 - 3.5i)T^{2} \)
13 \( 1 + (0.0948 + 0.354i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-4.85 + 4.85i)T - 17iT^{2} \)
19 \( 1 - 0.885iT - 19T^{2} \)
23 \( 1 + (3.43 - 0.920i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-4.96 - 8.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.994 - 1.72i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.65 - 1.65i)T + 37iT^{2} \)
41 \( 1 + (5.56 + 3.21i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-11.6 - 3.10i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-6.46 - 1.73i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.323 - 0.323i)T + 53iT^{2} \)
59 \( 1 + (-7.09 + 12.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.723 - 1.25i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.32 - 2.23i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 3.04iT - 71T^{2} \)
73 \( 1 + (9.32 - 9.32i)T - 73iT^{2} \)
79 \( 1 + (6.10 - 3.52i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.79 + 6.68i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 2.68T + 89T^{2} \)
97 \( 1 + (3.61 - 13.5i)T + (-84.0 - 48.5i)T^{2} \)
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

−10.56190797427630728759351545594, −9.938961132586465637199826151153, −9.192159066441571099191212660348, −8.293453621844209486643676785578, −7.26579878092269510933159489525, −5.71352178683339259229956071050, −4.92613626145986824580692666405, −4.25329066715486892149047217652, −3.37174512556865488648572088178, −0.999352133094250018266267359609, 2.12499414454464128363160166240, 2.90437438147750207155136099383, 4.08073996175292453023707360719, 5.79529847609124081028571545000, 6.02639673260747883161019150657, 7.53384193567856302618148984043, 8.313020562360785880847948097655, 9.052620701436920166225554573522, 10.11954888159373447806047480945, 11.57533069707178182257867958459

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