↑

Properties

Label 2-1638-91.81-c1-0-12
Degree $2$
Conductor $1638$
Sign $-0.955 - 0.294i$
Analytic cond. $13.0794$
Root an. cond. $3.61655$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (2.07 + 3.58i)5-s + (2.11 − 1.59i)7-s + 0.999·8-s − 4.14·10-s − 0.523·11-s + (−3.28 + 1.47i)13-s + (0.321 + 2.62i)14-s + (−0.5 + 0.866i)16-s + (−1.26 − 2.18i)17-s − 5.69·19-s + (2.07 − 3.58i)20-s + (0.261 − 0.453i)22-s + (−3.69 + 6.40i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.926 + 1.60i)5-s + (0.798 − 0.601i)7-s + 0.353·8-s − 1.30·10-s − 0.157·11-s + (−0.912 + 0.409i)13-s + (0.0859 + 0.701i)14-s + (−0.125 + 0.216i)16-s + (−0.305 − 0.529i)17-s − 1.30·19-s + (0.463 − 0.802i)20-s + (0.0557 − 0.0965i)22-s + (−0.770 + 1.33i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.955 - 0.294i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.955 - 0.294i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.206872604\)
\(L(\frac12)\) \(\approx\) \(1.206872604\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-2.11 + 1.59i)T \)
13 \( 1 + (3.28 - 1.47i)T \)
good5 \( 1 + (-2.07 - 3.58i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 0.523T + 11T^{2} \)
17 \( 1 + (1.26 + 2.18i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 5.69T + 19T^{2} \)
23 \( 1 + (3.69 - 6.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.54 - 2.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.17 - 3.76i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.83 + 4.90i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.33 + 4.03i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.81 - 8.34i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.58 - 9.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.00192 + 0.00332i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.05 - 7.02i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 6.01T + 61T^{2} \)
67 \( 1 - 3.23T + 67T^{2} \)
71 \( 1 + (3.98 - 6.90i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.99 + 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.15 - 2.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.08T + 83T^{2} \)
89 \( 1 + (5.42 - 9.39i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.31 - 2.28i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.788704943746448465461223460756, −9.035847075482463576994696369115, −7.88632224296560806756824809056, −7.24509259431585610851620222591, −6.72523882502093176397936482377, −5.87708989559713058471695415154, −4.99581601205103573581056289631, −3.91402411239432236253406644021, −2.58640986038864250223147618945, −1.73454853671844120090294571878, 0.48464629008360058992743318507, 1.95288942023364604942614852031, 2.30006224098633468228920528204, 4.17530875655603416835455684060, 4.81267382185722927463082615907, 5.53967927379897791142287324571, 6.45084821616929636080656162650, 7.920606028186996972615967641828, 8.548280664978265683399058022009, 8.812996470237687512104551139330

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