↑

Properties

Label 2-1305-1.1-c1-0-7
Degree 22
Conductor 13051305
Sign 11
Analytic cond. 10.420410.4204
Root an. cond. 3.228073.22807
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.431·2-s − 1.81·4-s + 5-s − 2.42·7-s − 1.64·8-s + 0.431·10-s − 0.00755·11-s − 2.73·13-s − 1.04·14-s + 2.91·16-s + 4.73·17-s + 2.74·19-s − 1.81·20-s − 0.00325·22-s + 3.68·23-s + 25-s − 1.17·26-s + 4.39·28-s − 29-s + 1.25·31-s + 4.55·32-s + 2.04·34-s − 2.42·35-s + 6.60·37-s + 1.18·38-s − 1.64·40-s + 0.160·41-s + ⋯
L(s)  = 1  + 0.305·2-s − 0.906·4-s + 0.447·5-s − 0.916·7-s − 0.581·8-s + 0.136·10-s − 0.00227·11-s − 0.757·13-s − 0.279·14-s + 0.729·16-s + 1.14·17-s + 0.629·19-s − 0.405·20-s − 0.000694·22-s + 0.769·23-s + 0.200·25-s − 0.231·26-s + 0.831·28-s − 0.185·29-s + 0.225·31-s + 0.804·32-s + 0.350·34-s − 0.409·35-s + 1.08·37-s + 0.192·38-s − 0.260·40-s + 0.0250·41-s + ⋯

Functional equation

Λ(s)=(1305s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1305s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13051305    =    32⋅5⋅293^{2} \cdot 5 \cdot 29
Sign: 11
Analytic conductor: 10.420410.4204
Root analytic conductor: 3.228073.22807
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1305, ( :1/2), 1)(2,\ 1305,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4183535561.418353556
L(12)L(\frac12) ≈\approx 1.4183535561.418353556
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
5 1−T 1 - T
29 1+T 1 + T
good2 1−0.431T+2T2 1 - 0.431T + 2T^{2}
7 1+2.42T+7T2 1 + 2.42T + 7T^{2}
11 1+0.00755T+11T2 1 + 0.00755T + 11T^{2}
13 1+2.73T+13T2 1 + 2.73T + 13T^{2}
17 1−4.73T+17T2 1 - 4.73T + 17T^{2}
19 1−2.74T+19T2 1 - 2.74T + 19T^{2}
23 1−3.68T+23T2 1 - 3.68T + 23T^{2}
31 1−1.25T+31T2 1 - 1.25T + 31T^{2}
37 1−6.60T+37T2 1 - 6.60T + 37T^{2}
41 1−0.160T+41T2 1 - 0.160T + 41T^{2}
43 1−11.0T+43T2 1 - 11.0T + 43T^{2}
47 1−10.3T+47T2 1 - 10.3T + 47T^{2}
53 1+11.1T+53T2 1 + 11.1T + 53T^{2}
59 1+8.85T+59T2 1 + 8.85T + 59T^{2}
61 1−13.5T+61T2 1 - 13.5T + 61T^{2}
67 1−4.69T+67T2 1 - 4.69T + 67T^{2}
71 1−1.48T+71T2 1 - 1.48T + 71T^{2}
73 1−2.75T+73T2 1 - 2.75T + 73T^{2}
79 1−2.72T+79T2 1 - 2.72T + 79T^{2}
83 1+7.81T+83T2 1 + 7.81T + 83T^{2}
89 1−7.59T+89T2 1 - 7.59T + 89T^{2}
97 1+7.02T+97T2 1 + 7.02T + 97T^{2}
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−9.579621732537984826438462580789, −9.150471074847902564009627814376, −8.013170310112582796775189720991, −7.20565407356210053359518570791, −6.09659681256263149858805649417, −5.46529441741522572718183172847, −4.58301396687217729650998958456, −3.50627935372303529769963027706, −2.69670009959084928639535592460, −0.847877574432523795244460143323, 0.847877574432523795244460143323, 2.69670009959084928639535592460, 3.50627935372303529769963027706, 4.58301396687217729650998958456, 5.46529441741522572718183172847, 6.09659681256263149858805649417, 7.20565407356210053359518570791, 8.013170310112582796775189720991, 9.150471074847902564009627814376, 9.579621732537984826438462580789

Graph of the ZZ-function along the critical line