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Properties

Label 2-175-7.6-c6-0-41
Degree 22
Conductor 175175
Sign 11
Analytic cond. 40.259440.2594
Root an. cond. 6.345036.34503
Motivic weight 66
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·2-s + 17·4-s + 343·7-s + 423·8-s + 729·9-s + 1.96e3·11-s − 3.08e3·14-s − 4.89e3·16-s − 6.56e3·18-s − 1.76e4·22-s + 2.27e4·23-s + 5.83e3·28-s − 2.12e4·29-s + 1.69e4·32-s + 1.23e4·36-s − 1.01e5·37-s + 1.26e5·43-s + 3.33e4·44-s − 2.04e5·46-s + 1.17e5·49-s − 5.03e4·53-s + 1.45e5·56-s + 1.90e5·58-s + 2.50e5·63-s + 1.60e5·64-s + 5.39e4·67-s − 2.42e5·71-s + ⋯
L(s)  = 1  − 9/8·2-s + 0.265·4-s + 7-s + 0.826·8-s + 9-s + 1.47·11-s − 9/8·14-s − 1.19·16-s − 9/8·18-s − 1.65·22-s + 1.86·23-s + 0.265·28-s − 0.870·29-s + 0.518·32-s + 0.265·36-s − 1.99·37-s + 1.59·43-s + 0.391·44-s − 2.10·46-s + 49-s − 0.338·53-s + 0.826·56-s + 0.978·58-s + 63-s + 0.612·64-s + 0.179·67-s − 0.677·71-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=( Λ(7−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 11
Analytic conductor: 40.259440.2594
Root analytic conductor: 6.345036.34503
Motivic weight: 66
Rational: yes
Arithmetic: yes
Character: χ175(76,⋅)\chi_{175} (76, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 175, ( :3), 1)(2,\ 175,\ (\ :3),\ 1)

Particular Values

L(72)L(\frac{7}{2}) ≈\approx 1.5054681531.505468153
L(12)L(\frac12) ≈\approx 1.5054681531.505468153
L(4)L(4) 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−p3T 1 - p^{3} T
good2 1+9T+p6T2 1 + 9 T + p^{6} T^{2}
3 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
11 1−1962T+p6T2 1 - 1962 T + p^{6} T^{2}
13 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
17 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
19 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
23 1−22734T+p6T2 1 - 22734 T + p^{6} T^{2}
29 1+21222T+p6T2 1 + 21222 T + p^{6} T^{2}
31 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
37 1+101194T+p6T2 1 + 101194 T + p^{6} T^{2}
41 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
43 1−126614T+p6T2 1 - 126614 T + p^{6} T^{2}
47 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
53 1+50346T+p6T2 1 + 50346 T + p^{6} T^{2}
59 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
61 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
67 1−53926T+p6T2 1 - 53926 T + p^{6} T^{2}
71 1+242478T+p6T2 1 + 242478 T + p^{6} T^{2}
73 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
79 1−929378T+p6T2 1 - 929378 T + p^{6} T^{2}
83 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
89 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
97 (1−p3T)(1+p3T) ( 1 - p^{3} T )( 1 + p^{3} T )
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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

−11.27458980478997798340008996115, −10.54036072421872383781365389031, −9.341515849022850901502772137087, −8.815265848002092986598830121936, −7.57620396673476033871019608336, −6.84022310351593936032661302339, −5.00859433059883282736261187573, −3.95719686563469584801833236872, −1.73276481640633712775694300103, −0.964393566757617540893754953678, 0.964393566757617540893754953678, 1.73276481640633712775694300103, 3.95719686563469584801833236872, 5.00859433059883282736261187573, 6.84022310351593936032661302339, 7.57620396673476033871019608336, 8.815265848002092986598830121936, 9.341515849022850901502772137087, 10.54036072421872383781365389031, 11.27458980478997798340008996115

Graph of the ZZ-function along the critical line