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Properties

Label 2-627-1.1-c1-0-7
Degree 22
Conductor 627627
Sign 11
Analytic cond. 5.006625.00662
Root an. cond. 2.237542.23754
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s + 2·7-s + 9-s + 11-s − 2·12-s − 13-s + 4·16-s + 3·17-s + 19-s + 2·21-s + 6·23-s − 5·25-s + 27-s − 4·28-s + 8·31-s + 33-s − 2·36-s + 2·37-s − 39-s + 6·41-s + 8·43-s − 2·44-s − 6·47-s + 4·48-s − 3·49-s + 3·51-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.577·12-s − 0.277·13-s + 16-s + 0.727·17-s + 0.229·19-s + 0.436·21-s + 1.25·23-s − 25-s + 0.192·27-s − 0.755·28-s + 1.43·31-s + 0.174·33-s − 1/3·36-s + 0.328·37-s − 0.160·39-s + 0.937·41-s + 1.21·43-s − 0.301·44-s − 0.875·47-s + 0.577·48-s − 3/7·49-s + 0.420·51-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: 11
Analytic conductor: 5.006625.00662
Root analytic conductor: 2.237542.23754
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 627, ( :1/2), 1)(2,\ 627,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6315590081.631559008
L(12)L(\frac12) ≈\approx 1.6315590081.631559008
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+pT2 1 + p T^{2}
5 1+pT2 1 + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1+6T+pT2 1 + 6 T + p T^{2}
53 1−9T+pT2 1 - 9 T + p T^{2}
59 1−3T+pT2 1 - 3 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1+10T+pT2 1 + 10 T + p T^{2}
71 1+3T+pT2 1 + 3 T + p T^{2}
73 1+4T+pT2 1 + 4 T + p T^{2}
79 1+13T+pT2 1 + 13 T + p T^{2}
83 1+3T+pT2 1 + 3 T + p T^{2}
89 1−15T+pT2 1 - 15 T + p T^{2}
97 1+10T+pT2 1 + 10 T + p T^{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

−10.38316548933611880460422425874, −9.595676059271080215098446016156, −8.885855868195721527212596861827, −8.047058109228960310569118011279, −7.38987844970363499694447516693, −5.92615105889325146373690957476, −4.88303727227954019406472250536, −4.10497206474885287550986629629, −2.89395743647375258219896560652, −1.21678682746529038884045056230, 1.21678682746529038884045056230, 2.89395743647375258219896560652, 4.10497206474885287550986629629, 4.88303727227954019406472250536, 5.92615105889325146373690957476, 7.38987844970363499694447516693, 8.047058109228960310569118011279, 8.885855868195721527212596861827, 9.595676059271080215098446016156, 10.38316548933611880460422425874

Graph of the ZZ-function along the critical line