↑

Properties

Label 2-627-1.1-c1-0-12
Degree 22
Conductor 627627
Sign −1-1
Analytic cond. 5.006625.00662
Root an. cond. 2.237542.23754
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.27·2-s − 3-s − 0.377·4-s − 1.37·5-s + 1.27·6-s + 0.651·7-s + 3.02·8-s + 9-s + 1.75·10-s + 11-s + 0.377·12-s − 4.27·13-s − 0.829·14-s + 1.37·15-s − 3.10·16-s + 7.33·17-s − 1.27·18-s + 19-s + 0.519·20-s − 0.651·21-s − 1.27·22-s − 1.20·23-s − 3.02·24-s − 3.10·25-s + 5.44·26-s − 27-s − 0.245·28-s + ⋯
L(s)  = 1  − 0.900·2-s − 0.577·3-s − 0.188·4-s − 0.615·5-s + 0.520·6-s + 0.246·7-s + 1.07·8-s + 0.333·9-s + 0.554·10-s + 0.301·11-s + 0.108·12-s − 1.18·13-s − 0.221·14-s + 0.355·15-s − 0.775·16-s + 1.77·17-s − 0.300·18-s + 0.229·19-s + 0.116·20-s − 0.142·21-s − 0.271·22-s − 0.251·23-s − 0.618·24-s − 0.620·25-s + 1.06·26-s − 0.192·27-s − 0.0464·28-s + ⋯

Functional equation

Λ(s)=(627s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(627s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 627627    =    3⋅11⋅193 \cdot 11 \cdot 19
Sign: −1-1
Analytic conductor: 5.006625.00662
Root analytic conductor: 2.237542.23754
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 627, ( :1/2), −1)(2,\ 627,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+T 1 + T
11 1−T 1 - T
19 1−T 1 - T
good2 1+1.27T+2T2 1 + 1.27T + 2T^{2}
5 1+1.37T+5T2 1 + 1.37T + 5T^{2}
7 1−0.651T+7T2 1 - 0.651T + 7T^{2}
13 1+4.27T+13T2 1 + 4.27T + 13T^{2}
17 1−7.33T+17T2 1 - 7.33T + 17T^{2}
23 1+1.20T+23T2 1 + 1.20T + 23T^{2}
29 1−1.10T+29T2 1 - 1.10T + 29T^{2}
31 1+4.19T+31T2 1 + 4.19T + 31T^{2}
37 1+0.245T+37T2 1 + 0.245T + 37T^{2}
41 1+7.23T+41T2 1 + 7.23T + 41T^{2}
43 1+7.67T+43T2 1 + 7.67T + 43T^{2}
47 1+3.82T+47T2 1 + 3.82T + 47T^{2}
53 1+7.47T+53T2 1 + 7.47T + 53T^{2}
59 1−6.85T+59T2 1 - 6.85T + 59T^{2}
61 1+3.37T+61T2 1 + 3.37T + 61T^{2}
67 1+9.57T+67T2 1 + 9.57T + 67T^{2}
71 1+0.576T+71T2 1 + 0.576T + 71T^{2}
73 1−5.33T+73T2 1 - 5.33T + 73T^{2}
79 1−6.23T+79T2 1 - 6.23T + 79T^{2}
83 1+4.07T+83T2 1 + 4.07T + 83T^{2}
89 1−6.46T+89T2 1 - 6.46T + 89T^{2}
97 1+15.9T+97T2 1 + 15.9T + 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

−9.924549471617796332262500994941, −9.583585427432778695523747435070, −8.253760316686257791040291356920, −7.73361933927410955078460037748, −6.90592032690430360733658389615, −5.46698421603451104779260461323, −4.68140921102196253909645667842, −3.50169197808424830648642498962, −1.53482211517777572812253890228, 0, 1.53482211517777572812253890228, 3.50169197808424830648642498962, 4.68140921102196253909645667842, 5.46698421603451104779260461323, 6.90592032690430360733658389615, 7.73361933927410955078460037748, 8.253760316686257791040291356920, 9.583585427432778695523747435070, 9.924549471617796332262500994941

Graph of the ZZ-function along the critical line