↑

Properties

Label 2-1260-28.27-c1-0-16
Degree $2$
Conductor $1260$
Sign $0.997 + 0.0771i$
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  + (0.102 − 1.41i)2-s + (−1.97 − 0.288i)4-s − i·5-s + (−0.178 − 2.63i)7-s + (−0.608 + 2.76i)8-s + (−1.41 − 0.102i)10-s + 5.22i·11-s + 4.52i·13-s + (−3.74 − 0.0184i)14-s + (3.83 + 1.14i)16-s + 6.70i·17-s + 2.81·19-s + (−0.288 + 1.97i)20-s + (7.37 + 0.534i)22-s − 0.858i·23-s + ⋯
L(s)  = 1  + (0.0722 − 0.997i)2-s + (−0.989 − 0.144i)4-s − 0.447i·5-s + (−0.0673 − 0.997i)7-s + (−0.215 + 0.976i)8-s + (−0.446 − 0.0323i)10-s + 1.57i·11-s + 1.25i·13-s + (−0.999 − 0.00493i)14-s + (0.958 + 0.285i)16-s + 1.62i·17-s + 0.646·19-s + (−0.0644 + 0.442i)20-s + (1.57 + 0.113i)22-s − 0.179i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.997 + 0.0771i$
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: \((2,\ 1260,\ (\ :1/2),\ 0.997 + 0.0771i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.208770525\)
\(L(\frac12)\) \(\approx\) \(1.208770525\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.102 + 1.41i)T \)
3 \( 1 \)
5 \( 1 + iT \)
7 \( 1 + (0.178 + 2.63i)T \)
good11 \( 1 - 5.22iT - 11T^{2} \)
13 \( 1 - 4.52iT - 13T^{2} \)
17 \( 1 - 6.70iT - 17T^{2} \)
19 \( 1 - 2.81T + 19T^{2} \)
23 \( 1 + 0.858iT - 23T^{2} \)
29 \( 1 + 6.47T + 29T^{2} \)
31 \( 1 - 2.60T + 31T^{2} \)
37 \( 1 - 2.13T + 37T^{2} \)
41 \( 1 - 8.71iT - 41T^{2} \)
43 \( 1 + 7.42iT - 43T^{2} \)
47 \( 1 - 9.82T + 47T^{2} \)
53 \( 1 + 3.69T + 53T^{2} \)
59 \( 1 + 4.27T + 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 - 4.52iT - 67T^{2} \)
71 \( 1 - 7.23iT - 71T^{2} \)
73 \( 1 - 9.24iT - 73T^{2} \)
79 \( 1 - 2.68iT - 79T^{2} \)
83 \( 1 - 16.2T + 83T^{2} \)
89 \( 1 - 8.53iT - 89T^{2} \)
97 \( 1 - 10.5iT - 97T^{2} \)
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

−9.734583275067334662594257515211, −9.219097735497375058629175644279, −8.157368509031191485255546799755, −7.35144155207902089400478789418, −6.34169193551903963308516793304, −5.07655472000800598595328459930, −4.24986922275899565068971523820, −3.78482833548462150793611484961, −2.15054605451249715314440952702, −1.32229293728490377202152071241, 0.54961278607871079833094207539, 2.83818603961341933917566398802, 3.44419704252452729428132305856, 4.96675166730519547669078668962, 5.69777783669948210637284963553, 6.14574267943048932951310756900, 7.38144180694784319412109424952, 7.88811027804460487363753290614, 8.902745615799658164250222827959, 9.325243034864486619317279468184

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