↑

Properties

Label 2-1638-91.74-c1-0-8
Degree $2$
Conductor $1638$
Sign $0.456 - 0.889i$
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  + 2-s + 4-s + (−0.114 − 0.197i)5-s + (−2.59 − 0.518i)7-s + 8-s + (−0.114 − 0.197i)10-s + (1.70 + 2.95i)11-s + (−1.62 + 3.21i)13-s + (−2.59 − 0.518i)14-s + 16-s + 5.41·17-s + (−3.17 + 5.49i)19-s + (−0.114 − 0.197i)20-s + (1.70 + 2.95i)22-s + 1.91·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.0509 − 0.0883i)5-s + (−0.980 − 0.195i)7-s + 0.353·8-s + (−0.0360 − 0.0624i)10-s + (0.514 + 0.890i)11-s + (−0.451 + 0.892i)13-s + (−0.693 − 0.138i)14-s + 0.250·16-s + 1.31·17-s + (−0.727 + 1.26i)19-s + (−0.0254 − 0.0441i)20-s + (0.363 + 0.629i)22-s + 0.400·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.456 - 0.889i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.456 - 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.218251289\)
\(L(\frac12)\) \(\approx\) \(2.218251289\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 + (2.59 + 0.518i)T \)
13 \( 1 + (1.62 - 3.21i)T \)
good5 \( 1 + (0.114 + 0.197i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.70 - 2.95i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 5.41T + 17T^{2} \)
19 \( 1 + (3.17 - 5.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.91T + 23T^{2} \)
29 \( 1 + (0.851 - 1.47i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.78 - 3.08i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.18T + 37T^{2} \)
41 \( 1 + (1.59 - 2.75i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.17 - 8.97i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.57 - 9.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.31 + 5.74i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 14.7T + 59T^{2} \)
61 \( 1 + (-3.35 + 5.81i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.58 + 9.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.0390 - 0.0677i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.31 - 10.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.811 - 1.40i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.70T + 83T^{2} \)
89 \( 1 + 17.5T + 89T^{2} \)
97 \( 1 + (-6.82 - 11.8i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.765824986710833999318095656263, −8.777743270209386265681996161261, −7.71415339926770284034417465906, −6.91912303981274044675316177655, −6.35610015930067187543201277951, −5.41258923876146309137903487740, −4.38140228635514737774091943422, −3.73616293621184387211646492150, −2.68603852968448934759036146485, −1.44766302529326424315718417404, 0.69514621353775475332790159498, 2.49989707604713457488617401286, 3.28321686047153876196926340480, 4.00965656389726406621917589645, 5.44635131554961176756929526675, 5.68606032540599926679158737898, 6.87954927849809901828575480446, 7.30965490557141334879998061750, 8.559902220494549614922403009068, 9.169277792631660645562253669372

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