↑

Properties

Label 2-740-740.739-c0-0-4
Degree 22
Conductor 740740
Sign 11
Analytic cond. 0.3693080.369308
Root an. cond. 0.6077070.607707
Motivic weight 00
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s + 8-s − 9-s + 10-s − 2·13-s + 16-s − 2·17-s − 18-s + 20-s + 25-s − 2·26-s + 32-s − 2·34-s − 36-s + 37-s + 40-s + 2·41-s − 45-s − 49-s + 50-s − 2·52-s + 64-s − 2·65-s − 2·68-s − 72-s + ⋯
L(s)  = 1  + 2-s + 4-s + 5-s + 8-s − 9-s + 10-s − 2·13-s + 16-s − 2·17-s − 18-s + 20-s + 25-s − 2·26-s + 32-s − 2·34-s − 36-s + 37-s + 40-s + 2·41-s − 45-s − 49-s + 50-s − 2·52-s + 64-s − 2·65-s − 2·68-s − 72-s + ⋯

Functional equation

Λ(s)=(740s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(740s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 740740    =    22⋅5⋅372^{2} \cdot 5 \cdot 37
Sign: 11
Analytic conductor: 0.3693080.369308
Root analytic conductor: 0.6077070.607707
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ740(739,⋅)\chi_{740} (739, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 740, ( :0), 1)(2,\ 740,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7708097711.770809771
L(12)L(\frac12) ≈\approx 1.7708097711.770809771
L(1)L(1) 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−T 1 - T
37 1−T 1 - T
good3 1+T2 1 + T^{2}
7 1+T2 1 + T^{2}
11 (1−T)(1+T) ( 1 - T )( 1 + T )
13 (1+T)2 ( 1 + T )^{2}
17 (1+T)2 ( 1 + T )^{2}
19 1+T2 1 + T^{2}
23 (1−T)(1+T) ( 1 - T )( 1 + T )
29 (1−T)(1+T) ( 1 - T )( 1 + T )
31 1+T2 1 + T^{2}
41 (1−T)2 ( 1 - T )^{2}
43 (1−T)(1+T) ( 1 - T )( 1 + T )
47 1+T2 1 + T^{2}
53 (1−T)(1+T) ( 1 - T )( 1 + T )
59 1+T2 1 + T^{2}
61 (1−T)(1+T) ( 1 - T )( 1 + T )
67 1+T2 1 + T^{2}
71 (1−T)(1+T) ( 1 - T )( 1 + T )
73 (1−T)(1+T) ( 1 - T )( 1 + T )
79 1+T2 1 + T^{2}
83 1+T2 1 + T^{2}
89 (1−T)(1+T) ( 1 - T )( 1 + T )
97 (1−T)2 ( 1 - 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

−10.80821644897585403714162290343, −9.780213099441346580515854375829, −9.026933799111392309276303617302, −7.77567985936988259385724288307, −6.79294902490755101631249068117, −6.06124431521073214267295899939, −5.13488793212626901170948348056, −4.41645981063411314247634537539, −2.72816018386619166937329311600, −2.24576785584387077945318324787, 2.24576785584387077945318324787, 2.72816018386619166937329311600, 4.41645981063411314247634537539, 5.13488793212626901170948348056, 6.06124431521073214267295899939, 6.79294902490755101631249068117, 7.77567985936988259385724288307, 9.026933799111392309276303617302, 9.780213099441346580515854375829, 10.80821644897585403714162290343

Graph of the ZZ-function along the critical line