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Properties

Label 2-175-7.6-c2-0-16
Degree 22
Conductor 175175
Sign 11
Analytic cond. 4.768404.76840
Root an. cond. 2.183662.18366
Motivic weight 22
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 5·4-s + 7·7-s + 3·8-s + 9·9-s − 6·11-s + 21·14-s − 11·16-s + 27·18-s − 18·22-s − 18·23-s + 35·28-s − 54·29-s − 45·32-s + 45·36-s + 38·37-s − 58·43-s − 30·44-s − 54·46-s + 49·49-s + 6·53-s + 21·56-s − 162·58-s + 63·63-s − 91·64-s + 118·67-s + 114·71-s + ⋯
L(s)  = 1  + 3/2·2-s + 5/4·4-s + 7-s + 3/8·8-s + 9-s − 0.545·11-s + 3/2·14-s − 0.687·16-s + 3/2·18-s − 0.818·22-s − 0.782·23-s + 5/4·28-s − 1.86·29-s − 1.40·32-s + 5/4·36-s + 1.02·37-s − 1.34·43-s − 0.681·44-s − 1.17·46-s + 49-s + 6/53·53-s + 3/8·56-s − 2.79·58-s + 63-s − 1.42·64-s + 1.76·67-s + 1.60·71-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 11
Analytic conductor: 4.768404.76840
Root analytic conductor: 2.183662.18366
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: χ175(76,⋅)\chi_{175} (76, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 175, ( :1), 1)(2,\ 175,\ (\ :1),\ 1)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 3.3406448293.340644829
L(12)L(\frac12) ≈\approx 3.3406448293.340644829
L(2)L(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)
bad5 1 1
7 1−pT 1 - p T
good2 1−3T+p2T2 1 - 3 T + p^{2} T^{2}
3 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
11 1+6T+p2T2 1 + 6 T + p^{2} T^{2}
13 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
17 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
19 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
23 1+18T+p2T2 1 + 18 T + p^{2} T^{2}
29 1+54T+p2T2 1 + 54 T + p^{2} T^{2}
31 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
37 1−38T+p2T2 1 - 38 T + p^{2} T^{2}
41 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
43 1+58T+p2T2 1 + 58 T + p^{2} T^{2}
47 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
53 1−6T+p2T2 1 - 6 T + p^{2} T^{2}
59 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
61 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
67 1−118T+p2T2 1 - 118 T + p^{2} T^{2}
71 1−114T+p2T2 1 - 114 T + p^{2} T^{2}
73 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
79 1+94T+p2T2 1 + 94 T + p^{2} T^{2}
83 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
89 (1−pT)(1+pT) ( 1 - p T )( 1 + p T )
97 (1−pT)(1+pT) ( 1 - p T )( 1 + p 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

−12.73499309097954401241261364186, −11.72114946429536415960283641224, −10.90396618488062604506571068582, −9.634744201767806553925033946699, −8.094351705474056676734448544284, −7.03718216062352377145727309085, −5.70021710145192853096878990643, −4.75622814773392679209724927762, −3.78207417059432049215665954029, −2.05341769902145376345960214580, 2.05341769902145376345960214580, 3.78207417059432049215665954029, 4.75622814773392679209724927762, 5.70021710145192853096878990643, 7.03718216062352377145727309085, 8.094351705474056676734448544284, 9.634744201767806553925033946699, 10.90396618488062604506571068582, 11.72114946429536415960283641224, 12.73499309097954401241261364186

Graph of the ZZ-function along the critical line