↑

Properties

Label 2-950-1.1-c1-0-3
Degree 22
Conductor 950950
Sign 11
Analytic cond. 7.585787.58578
Root an. cond. 2.754232.75423
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s − 2·9-s − 12-s + 13-s − 14-s + 16-s + 3·17-s + 2·18-s + 19-s − 21-s − 3·23-s + 24-s − 26-s + 5·27-s + 28-s − 3·29-s + 2·31-s − 32-s − 3·34-s − 2·36-s + 10·37-s − 38-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.471·18-s + 0.229·19-s − 0.218·21-s − 0.625·23-s + 0.204·24-s − 0.196·26-s + 0.962·27-s + 0.188·28-s − 0.557·29-s + 0.359·31-s − 0.176·32-s − 0.514·34-s − 1/3·36-s + 1.64·37-s − 0.162·38-s + ⋯

Functional equation

Λ(s)=(950s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(950s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 950950    =    2⋅52⋅192 \cdot 5^{2} \cdot 19
Sign: 11
Analytic conductor: 7.585787.58578
Root analytic conductor: 2.754232.75423
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 950, ( :1/2), 1)(2,\ 950,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.88211212330.8821121233
L(12)L(\frac12) ≈\approx 0.88211212330.8821121233
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)
bad2 1+T 1 + T
5 1 1
19 1−T 1 - T
good3 1+T+pT2 1 + T + p T^{2}
7 1−T+pT2 1 - T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
23 1+3T+pT2 1 + 3 T + p T^{2}
29 1+3T+pT2 1 + 3 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+2T+pT2 1 + 2 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+3T+pT2 1 + 3 T + p T^{2}
59 1−3T+pT2 1 - 3 T + p T^{2}
61 1−8T+pT2 1 - 8 T + p T^{2}
67 1−7T+pT2 1 - 7 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1−13T+pT2 1 - 13 T + p T^{2}
79 1−14T+pT2 1 - 14 T + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−10T+pT2 1 - 10 T + p T^{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

−9.983035167615231513039157588703, −9.303912250997036448752264323167, −8.237637220939037195909198183700, −7.77982432388455298937264149525, −6.57249513174536622080526940747, −5.85301595783600076781492520747, −4.98057051522020806813162729411, −3.61241990046837984333714647492, −2.34024048259330359076586962577, −0.850809003224210732230405050623, 0.850809003224210732230405050623, 2.34024048259330359076586962577, 3.61241990046837984333714647492, 4.98057051522020806813162729411, 5.85301595783600076781492520747, 6.57249513174536622080526940747, 7.77982432388455298937264149525, 8.237637220939037195909198183700, 9.303912250997036448752264323167, 9.983035167615231513039157588703

Graph of the ZZ-function along the critical line