↑

Properties

Label 2-175-7.6-c8-0-77
Degree 22
Conductor 175175
Sign 11
Analytic cond. 71.291271.2912
Root an. cond. 8.443418.44341
Motivic weight 88
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 31·2-s + 705·4-s − 2.40e3·7-s + 1.39e4·8-s + 6.56e3·9-s + 1.31e4·11-s − 7.44e4·14-s + 2.51e5·16-s + 2.03e5·18-s + 4.07e5·22-s + 2.09e4·23-s − 1.69e6·28-s + 1.08e5·29-s + 4.21e6·32-s + 4.62e6·36-s + 2.07e6·37-s + 6.72e6·43-s + 9.27e6·44-s + 6.48e5·46-s + 5.76e6·49-s − 1.53e7·53-s − 3.34e7·56-s + 3.35e6·58-s − 1.57e7·63-s + 6.65e7·64-s + 1.58e7·67-s − 4.23e7·71-s + ⋯
L(s)  = 1  + 1.93·2-s + 2.75·4-s − 7-s + 3.39·8-s + 9-s + 0.898·11-s − 1.93·14-s + 3.83·16-s + 1.93·18-s + 1.74·22-s + 0.0747·23-s − 2.75·28-s + 0.152·29-s + 4.02·32-s + 2.75·36-s + 1.10·37-s + 1.96·43-s + 2.47·44-s + 0.144·46-s + 49-s − 1.94·53-s − 3.39·56-s + 0.296·58-s − 63-s + 3.96·64-s + 0.786·67-s − 1.66·71-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=( Λ(9−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+4) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 11
Analytic conductor: 71.291271.2912
Root analytic conductor: 8.443418.44341
Motivic weight: 88
Rational: yes
Arithmetic: yes
Character: χ175(76,⋅)\chi_{175} (76, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 175, ( :4), 1)(2,\ 175,\ (\ :4),\ 1)

Particular Values

L(92)L(\frac{9}{2}) ≈\approx 8.6890818828.689081882
L(12)L(\frac12) ≈\approx 8.6890818828.689081882
L(5)L(5) 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)
bad5 1 1
7 1+p4T 1 + p^{4} T
good2 1−31T+p8T2 1 - 31 T + p^{8} T^{2}
3 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
11 1−13154T+p8T2 1 - 13154 T + p^{8} T^{2}
13 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
17 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
19 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
23 1−20926T+p8T2 1 - 20926 T + p^{8} T^{2}
29 1−108194T+p8T2 1 - 108194 T + p^{8} T^{2}
31 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
37 1−2073886T+p8T2 1 - 2073886 T + p^{8} T^{2}
41 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
43 1−6726046T+p8T2 1 - 6726046 T + p^{8} T^{2}
47 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
53 1+15377762T+p8T2 1 + 15377762 T + p^{8} T^{2}
59 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
61 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
67 1−15839326T+p8T2 1 - 15839326 T + p^{8} T^{2}
71 1+42331966T+p8T2 1 + 42331966 T + p^{8} T^{2}
73 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
79 1+64606846T+p8T2 1 + 64606846 T + p^{8} T^{2}
83 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
89 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
97 (1−p4T)(1+p4T) ( 1 - p^{4} T )( 1 + p^{4} T )
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

−11.62360575641308332786420220942, −10.57156358883316207945803237264, −9.483441745867234965478944743745, −7.51274088293784108999009292991, −6.64463201012051924082240774883, −5.91126240349294564794769788184, −4.52116871855317945142999139154, −3.78357905619237699182705988646, −2.67909132520827277110527058471, −1.30334214930380591256777708099, 1.30334214930380591256777708099, 2.67909132520827277110527058471, 3.78357905619237699182705988646, 4.52116871855317945142999139154, 5.91126240349294564794769788184, 6.64463201012051924082240774883, 7.51274088293784108999009292991, 9.483441745867234965478944743745, 10.57156358883316207945803237264, 11.62360575641308332786420220942

Graph of the ZZ-function along the critical line