↑

Properties

Label 2-1260-28.27-c1-0-74
Degree $2$
Conductor $1260$
Sign $-0.812 + 0.582i$
Analytic cond. $10.0611$
Root an. cond. $3.17193$
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  + (0.947 − 1.04i)2-s + (−0.204 − 1.98i)4-s − i·5-s + (2.29 − 1.31i)7-s + (−2.28 − 1.66i)8-s + (−1.04 − 0.947i)10-s − 0.477i·11-s − 2.96i·13-s + (0.796 − 3.65i)14-s + (−3.91 + 0.814i)16-s + 3.83i·17-s + 5.31·19-s + (−1.98 + 0.204i)20-s + (−0.500 − 0.452i)22-s − 7.60i·23-s + ⋯
L(s)  = 1  + (0.669 − 0.742i)2-s + (−0.102 − 0.994i)4-s − 0.447i·5-s + (0.868 − 0.496i)7-s + (−0.807 − 0.590i)8-s + (−0.332 − 0.299i)10-s − 0.143i·11-s − 0.821i·13-s + (0.212 − 0.977i)14-s + (−0.979 + 0.203i)16-s + 0.929i·17-s + 1.21·19-s + (−0.444 + 0.0457i)20-s + (−0.106 − 0.0963i)22-s − 1.58i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.812 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.812 + 0.582i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ -0.812 + 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.341449765\)
\(L(\frac12)\) \(\approx\) \(2.341449765\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.947 + 1.04i)T \)
3 \( 1 \)
5 \( 1 + iT \)
7 \( 1 + (-2.29 + 1.31i)T \)
good11 \( 1 + 0.477iT - 11T^{2} \)
13 \( 1 + 2.96iT - 13T^{2} \)
17 \( 1 - 3.83iT - 17T^{2} \)
19 \( 1 - 5.31T + 19T^{2} \)
23 \( 1 + 7.60iT - 23T^{2} \)
29 \( 1 + 6.17T + 29T^{2} \)
31 \( 1 - 3.38T + 31T^{2} \)
37 \( 1 + 8.62T + 37T^{2} \)
41 \( 1 + 1.01iT - 41T^{2} \)
43 \( 1 - 6.85iT - 43T^{2} \)
47 \( 1 + 6.21T + 47T^{2} \)
53 \( 1 - 9.30T + 53T^{2} \)
59 \( 1 - 4.88T + 59T^{2} \)
61 \( 1 - 4.75iT - 61T^{2} \)
67 \( 1 + 1.30iT - 67T^{2} \)
71 \( 1 + 9.18iT - 71T^{2} \)
73 \( 1 + 4.49iT - 73T^{2} \)
79 \( 1 + 8.80iT - 79T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 - 13.1iT - 89T^{2} \)
97 \( 1 + 1.60iT - 97T^{2} \)
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

−9.554909870900612311103929192628, −8.575926101345314569721162871097, −7.86643585474287998554021200176, −6.73537271327176269848038439994, −5.66559088782973175927866223689, −5.01058141788821154008877652117, −4.15091959760602348938001453171, −3.21924333972862801957377753083, −1.91513878378910681799511781183, −0.813144125612419738781755113393, 1.91068930049639494765294222666, 3.11579291028749059712155878224, 4.07035409933140851797483255638, 5.23045104704533005359191940677, 5.55288344482460288030940668957, 6.90988657619961191678851261780, 7.31369728212946396481995432423, 8.193582089055735260451289330890, 9.095447595304270868394156419885, 9.766661208767502129841301423742

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