↑

Properties

Label 2-1260-28.27-c1-0-66
Degree $2$
Conductor $1260$
Sign $-0.355 + 0.934i$
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  + (−1.10 + 0.887i)2-s + (0.426 − 1.95i)4-s − i·5-s + (0.391 − 2.61i)7-s + (1.26 + 2.53i)8-s + (0.887 + 1.10i)10-s − 0.770i·11-s − 5.60i·13-s + (1.88 + 3.22i)14-s + (−3.63 − 1.66i)16-s + 0.503i·17-s + 1.63·19-s + (−1.95 − 0.426i)20-s + (0.683 + 0.848i)22-s + 1.42i·23-s + ⋯
L(s)  = 1  + (−0.778 + 0.627i)2-s + (0.213 − 0.977i)4-s − 0.447i·5-s + (0.148 − 0.988i)7-s + (0.446 + 0.894i)8-s + (0.280 + 0.348i)10-s − 0.232i·11-s − 1.55i·13-s + (0.504 + 0.863i)14-s + (−0.909 − 0.416i)16-s + 0.122i·17-s + 0.375·19-s + (−0.436 − 0.0953i)20-s + (0.145 + 0.180i)22-s + 0.297i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.355 + 0.934i$
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: \((2,\ 1260,\ (\ :1/2),\ -0.355 + 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7473104447\)
\(L(\frac12)\) \(\approx\) \(0.7473104447\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.10 - 0.887i)T \)
3 \( 1 \)
5 \( 1 + iT \)
7 \( 1 + (-0.391 + 2.61i)T \)
good11 \( 1 + 0.770iT - 11T^{2} \)
13 \( 1 + 5.60iT - 13T^{2} \)
17 \( 1 - 0.503iT - 17T^{2} \)
19 \( 1 - 1.63T + 19T^{2} \)
23 \( 1 - 1.42iT - 23T^{2} \)
29 \( 1 + 5.03T + 29T^{2} \)
31 \( 1 + 8.23T + 31T^{2} \)
37 \( 1 - 10.1T + 37T^{2} \)
41 \( 1 + 5.07iT - 41T^{2} \)
43 \( 1 - 9.06iT - 43T^{2} \)
47 \( 1 + 4.64T + 47T^{2} \)
53 \( 1 + 0.455T + 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 + 3.32iT - 61T^{2} \)
67 \( 1 - 8.70iT - 67T^{2} \)
71 \( 1 + 10.8iT - 71T^{2} \)
73 \( 1 - 2.29iT - 73T^{2} \)
79 \( 1 + 2.56iT - 79T^{2} \)
83 \( 1 + 13.9T + 83T^{2} \)
89 \( 1 + 2.98iT - 89T^{2} \)
97 \( 1 + 15.5iT - 97T^{2} \)
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

−9.467518292688418538725397053764, −8.497325930289528444839366501823, −7.70650423348464568330880808399, −7.35572785889526016922377596901, −6.08733229489478423127072242975, −5.44730975972034635518093148382, −4.45966093241590432151323141853, −3.19773250716726127003610201420, −1.54644250730831161165899076144, −0.41172384659962475236013019822, 1.70605345317545136382271081494, 2.48937294032793724057958099563, 3.61916320645312621569521749742, 4.63789811442500175250515605889, 5.92078230739944663804276643560, 6.86207745252915784991340665171, 7.58339844141764894483694354662, 8.530800445258446211441292954921, 9.351935656196986493277645340307, 9.628631970642491797748026773684

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