Properties

Label 2-1260-28.27-c1-0-28
Degree $2$
Conductor $1260$
Sign $-0.115 - 0.993i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.39 + 0.238i)2-s + (1.88 + 0.664i)4-s + i·5-s + (−2.37 + 1.16i)7-s + (2.47 + 1.37i)8-s + (−0.238 + 1.39i)10-s + 4.86i·11-s − 3.63i·13-s + (−3.59 + 1.05i)14-s + (3.11 + 2.50i)16-s + 4.47i·17-s + 2.70·19-s + (−0.664 + 1.88i)20-s + (−1.16 + 6.78i)22-s − 1.68i·23-s + ⋯
L(s)  = 1  + (0.985 + 0.168i)2-s + (0.943 + 0.332i)4-s + 0.447i·5-s + (−0.898 + 0.439i)7-s + (0.873 + 0.486i)8-s + (−0.0754 + 0.440i)10-s + 1.46i·11-s − 1.00i·13-s + (−0.959 + 0.281i)14-s + (0.779 + 0.627i)16-s + 1.08i·17-s + 0.621·19-s + (−0.148 + 0.421i)20-s + (−0.247 + 1.44i)22-s − 0.351i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.115 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.115 - 0.993i$
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: \((2,\ 1260,\ (\ :1/2),\ -0.115 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.669372630\)
\(L(\frac12)\) \(\approx\) \(2.669372630\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 - 0.238i)T \)
3 \( 1 \)
5 \( 1 - iT \)
7 \( 1 + (2.37 - 1.16i)T \)
good11 \( 1 - 4.86iT - 11T^{2} \)
13 \( 1 + 3.63iT - 13T^{2} \)
17 \( 1 - 4.47iT - 17T^{2} \)
19 \( 1 - 2.70T + 19T^{2} \)
23 \( 1 + 1.68iT - 23T^{2} \)
29 \( 1 + 8.31T + 29T^{2} \)
31 \( 1 + 5.47T + 31T^{2} \)
37 \( 1 - 7.07T + 37T^{2} \)
41 \( 1 - 11.5iT - 41T^{2} \)
43 \( 1 - 7.86iT - 43T^{2} \)
47 \( 1 - 4.75T + 47T^{2} \)
53 \( 1 - 10.1T + 53T^{2} \)
59 \( 1 + 2.97T + 59T^{2} \)
61 \( 1 - 1.18iT - 61T^{2} \)
67 \( 1 + 13.1iT - 67T^{2} \)
71 \( 1 + 14.8iT - 71T^{2} \)
73 \( 1 + 1.40iT - 73T^{2} \)
79 \( 1 + 1.01iT - 79T^{2} \)
83 \( 1 - 8.22T + 83T^{2} \)
89 \( 1 + 9.91iT - 89T^{2} \)
97 \( 1 - 13.0iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.02043933432876785156536032195, −9.251205986829715216965905416797, −7.86133579602696411106033233295, −7.39609419514129928149936360248, −6.36872770149563973259543717048, −5.82839106784381104598049096684, −4.81715037228498155526635423382, −3.77280694776644080372715407739, −2.95098852433856239285052786197, −1.92893408703537665012021644186, 0.793440863809752224318764430463, 2.35965208929934620948642652453, 3.53473178357360514051159530525, 4.03947675012480383564888478118, 5.40554173375251137406770693173, 5.82149053663410657111735792712, 6.98900421727204078207262118618, 7.43890213018496066652197950755, 8.878787646731997003885981525037, 9.447661550895260483332883091352

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