↑

Properties

Label 2-14-1.1-c1-0-0
Degree 22
Conductor 1414
Sign 11
Analytic cond. 0.1117900.111790
Root an. cond. 0.3343500.334350
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s + 2·6-s + 7-s − 8-s + 9-s − 2·12-s − 4·13-s − 14-s + 16-s + 6·17-s − 18-s + 2·19-s − 2·21-s + 2·24-s − 5·25-s + 4·26-s + 4·27-s + 28-s − 6·29-s − 4·31-s − 32-s − 6·34-s + 36-s + 2·37-s − 2·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.577·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.458·19-s − 0.436·21-s + 0.408·24-s − 25-s + 0.784·26-s + 0.769·27-s + 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s + 0.328·37-s − 0.324·38-s + ⋯

Functional equation

Λ(s)=(14s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(14s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1414    =    2⋅72 \cdot 7
Sign: 11
Analytic conductor: 0.1117900.111790
Root analytic conductor: 0.3343500.334350
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 14, ( :1/2), 1)(2,\ 14,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.33022365930.3302236593
L(12)L(\frac12) ≈\approx 0.33022365930.3302236593
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
7 1−T 1 - T
good3 1+2T+pT2 1 + 2 T + p T^{2}
5 1+pT2 1 + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+4T+pT2 1 + 4 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+12T+pT2 1 + 12 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1−8T+pT2 1 - 8 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1−8T+pT2 1 - 8 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

−19.58002701984274477714555784487, −18.21205413166163302504660601599, −17.21285321618276241317782960612, −16.33708100139526708806159182558, −14.60777306567129604319700846806, −12.30526099005271094386573698884, −11.23136141438460806904855801814, −9.765547119459919407856461234632, −7.57571100088867902110310233811, −5.57928681742950427486583645839, 5.57928681742950427486583645839, 7.57571100088867902110310233811, 9.765547119459919407856461234632, 11.23136141438460806904855801814, 12.30526099005271094386573698884, 14.60777306567129604319700846806, 16.33708100139526708806159182558, 17.21285321618276241317782960612, 18.21205413166163302504660601599, 19.58002701984274477714555784487

Graph of the ZZ-function along the critical line