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Properties

Label 2-627-1.1-c1-0-25
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.760·2-s + 3-s − 1.42·4-s + 1.94·5-s − 0.760·6-s − 5.18·7-s + 2.60·8-s + 9-s − 1.47·10-s + 11-s − 1.42·12-s − 4.23·13-s + 3.94·14-s + 1.94·15-s + 0.861·16-s − 2.28·17-s − 0.760·18-s − 19-s − 2.76·20-s − 5.18·21-s − 0.760·22-s − 5·23-s + 2.60·24-s − 1.22·25-s + 3.22·26-s + 27-s + 7.36·28-s + ⋯
L(s)  = 1  − 0.538·2-s + 0.577·3-s − 0.710·4-s + 0.868·5-s − 0.310·6-s − 1.95·7-s + 0.920·8-s + 0.333·9-s − 0.467·10-s + 0.301·11-s − 0.410·12-s − 1.17·13-s + 1.05·14-s + 0.501·15-s + 0.215·16-s − 0.553·17-s − 0.179·18-s − 0.229·19-s − 0.617·20-s − 1.13·21-s − 0.162·22-s − 1.04·23-s + 0.531·24-s − 0.245·25-s + 0.632·26-s + 0.192·27-s + 1.39·28-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+0.760T+2T2 1 + 0.760T + 2T^{2}
5 1−1.94T+5T2 1 - 1.94T + 5T^{2}
7 1+5.18T+7T2 1 + 5.18T + 7T^{2}
13 1+4.23T+13T2 1 + 4.23T + 13T^{2}
17 1+2.28T+17T2 1 + 2.28T + 17T^{2}
23 1+5T+23T2 1 + 5T + 23T^{2}
29 1+5.13T+29T2 1 + 5.13T + 29T^{2}
31 1−6.66T+31T2 1 - 6.66T + 31T^{2}
37 1+9.72T+37T2 1 + 9.72T + 37T^{2}
41 1+0.239T+41T2 1 + 0.239T + 41T^{2}
43 1+0.578T+43T2 1 + 0.578T + 43T^{2}
47 1−3.12T+47T2 1 - 3.12T + 47T^{2}
53 1+8.82T+53T2 1 + 8.82T + 53T^{2}
59 1−9.54T+59T2 1 - 9.54T + 59T^{2}
61 1+9.46T+61T2 1 + 9.46T + 61T^{2}
67 1+10.9T+67T2 1 + 10.9T + 67T^{2}
71 1+7.23T+71T2 1 + 7.23T + 71T^{2}
73 1+5.12T+73T2 1 + 5.12T + 73T^{2}
79 1−7.03T+79T2 1 - 7.03T + 79T^{2}
83 1−13.3T+83T2 1 - 13.3T + 83T^{2}
89 1+6.66T+89T2 1 + 6.66T + 89T^{2}
97 1+8.74T+97T2 1 + 8.74T + 97T^{2}
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   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

−9.945960558921735708211469519964, −9.380918201746149763267946659055, −8.797656632967591469260696952254, −7.58100862009213872539159022626, −6.66591394883088851365464324913, −5.75014661566536491598819371277, −4.40703345469512683488161062797, −3.31743313767316729267868515798, −2.07609450278487697987943329898, 0, 2.07609450278487697987943329898, 3.31743313767316729267868515798, 4.40703345469512683488161062797, 5.75014661566536491598819371277, 6.66591394883088851365464324913, 7.58100862009213872539159022626, 8.797656632967591469260696952254, 9.380918201746149763267946659055, 9.945960558921735708211469519964

Graph of the ZZ-function along the critical line