↑

Properties

Label 2-1305-1.1-c1-0-4
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  − 1.79·2-s + 1.20·4-s − 5-s + 7-s + 1.41·8-s + 1.79·10-s − 5·11-s − 4.58·13-s − 1.79·14-s − 4.95·16-s + 3·17-s + 3.58·19-s − 1.20·20-s + 8.95·22-s + 4·23-s + 25-s + 8.20·26-s + 1.20·28-s − 29-s + 4·31-s + 6.04·32-s − 5.37·34-s − 35-s − 4·37-s − 6.41·38-s − 1.41·40-s + 9.16·41-s + ⋯
L(s)  = 1  − 1.26·2-s + 0.604·4-s − 0.447·5-s + 0.377·7-s + 0.501·8-s + 0.566·10-s − 1.50·11-s − 1.27·13-s − 0.478·14-s − 1.23·16-s + 0.727·17-s + 0.821·19-s − 0.270·20-s + 1.90·22-s + 0.834·23-s + 0.200·25-s + 1.60·26-s + 0.228·28-s − 0.185·29-s + 0.718·31-s + 1.06·32-s − 0.921·34-s − 0.169·35-s − 0.657·37-s − 1.04·38-s − 0.224·40-s + 1.43·41-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 0.57782138040.5778213804
L(12)L(\frac12) ≈\approx 0.57782138040.5778213804
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+1.79T+2T2 1 + 1.79T + 2T^{2}
7 1−T+7T2 1 - T + 7T^{2}
11 1+5T+11T2 1 + 5T + 11T^{2}
13 1+4.58T+13T2 1 + 4.58T + 13T^{2}
17 1−3T+17T2 1 - 3T + 17T^{2}
19 1−3.58T+19T2 1 - 3.58T + 19T^{2}
23 1−4T+23T2 1 - 4T + 23T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1+4T+37T2 1 + 4T + 37T^{2}
41 1−9.16T+41T2 1 - 9.16T + 41T^{2}
43 1+9.58T+43T2 1 + 9.58T + 43T^{2}
47 1+10.5T+47T2 1 + 10.5T + 47T^{2}
53 1+0.417T+53T2 1 + 0.417T + 53T^{2}
59 1−7.58T+59T2 1 - 7.58T + 59T^{2}
61 1−12.7T+61T2 1 - 12.7T + 61T^{2}
67 1+4.16T+67T2 1 + 4.16T + 67T^{2}
71 1−9.58T+71T2 1 - 9.58T + 71T^{2}
73 1−4T+73T2 1 - 4T + 73T^{2}
79 1−7.58T+79T2 1 - 7.58T + 79T^{2}
83 1−11.5T+83T2 1 - 11.5T + 83T^{2}
89 1+1.41T+89T2 1 + 1.41T + 89T^{2}
97 1−11.5T+97T2 1 - 11.5T + 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

−9.859305826106954012877599441931, −8.767952699352663135566731424857, −7.903555847505768846152238391293, −7.66886914070631679152925175304, −6.79452836761544173571285866947, −5.19039359417866767465910634504, −4.84005163502576090222198072988, −3.25117926876776484171041075535, −2.14483649046238086318972586305, −0.66149352268979187435367943342, 0.66149352268979187435367943342, 2.14483649046238086318972586305, 3.25117926876776484171041075535, 4.84005163502576090222198072988, 5.19039359417866767465910634504, 6.79452836761544173571285866947, 7.66886914070631679152925175304, 7.903555847505768846152238391293, 8.767952699352663135566731424857, 9.859305826106954012877599441931

Graph of the ZZ-function along the critical line