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Properties

Label 2-175-7.6-c4-0-16
Degree 22
Conductor 175175
Sign 11
Analytic cond. 18.089718.0897
Root an. cond. 4.253204.25320
Motivic weight 44
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 15·4-s − 49·7-s + 31·8-s + 81·9-s − 206·11-s + 49·14-s + 209·16-s − 81·18-s + 206·22-s + 734·23-s + 735·28-s + 1.23e3·29-s − 705·32-s − 1.21e3·36-s + 1.29e3·37-s + 334·43-s + 3.09e3·44-s − 734·46-s + 2.40e3·49-s + 5.58e3·53-s − 1.51e3·56-s − 1.23e3·58-s − 3.96e3·63-s − 2.63e3·64-s − 4.94e3·67-s + 2.91e3·71-s + ⋯
L(s)  = 1  − 1/4·2-s − 0.937·4-s − 7-s + 0.484·8-s + 9-s − 1.70·11-s + 1/4·14-s + 0.816·16-s − 1/4·18-s + 0.425·22-s + 1.38·23-s + 0.937·28-s + 1.46·29-s − 0.688·32-s − 0.937·36-s + 0.945·37-s + 0.180·43-s + 1.59·44-s − 0.346·46-s + 49-s + 1.98·53-s − 0.484·56-s − 0.366·58-s − 63-s − 0.644·64-s − 1.10·67-s + 0.578·71-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=( Λ(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 11
Analytic conductor: 18.089718.0897
Root analytic conductor: 4.253204.25320
Motivic weight: 44
Rational: yes
Arithmetic: yes
Character: χ175(76,⋅)\chi_{175} (76, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 175, ( :2), 1)(2,\ 175,\ (\ :2),\ 1)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.98673610740.9867361074
L(12)L(\frac12) ≈\approx 0.98673610740.9867361074
L(3)L(3) 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+p2T 1 + p^{2} T
good2 1+T+p4T2 1 + T + p^{4} T^{2}
3 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
11 1+206T+p4T2 1 + 206 T + p^{4} T^{2}
13 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
17 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
19 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
23 1−734T+p4T2 1 - 734 T + p^{4} T^{2}
29 1−1234T+p4T2 1 - 1234 T + p^{4} T^{2}
31 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
37 1−1294T+p4T2 1 - 1294 T + p^{4} T^{2}
41 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
43 1−334T+p4T2 1 - 334 T + p^{4} T^{2}
47 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
53 1−5582T+p4T2 1 - 5582 T + p^{4} T^{2}
59 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
61 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
67 1+4946T+p4T2 1 + 4946 T + p^{4} T^{2}
71 1−2914T+p4T2 1 - 2914 T + p^{4} T^{2}
73 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
79 1+3646T+p4T2 1 + 3646 T + p^{4} T^{2}
83 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
89 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
97 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} 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

−12.37396704661604279665517269154, −10.60038750242078630535181495454, −10.05956102446464702736683688807, −9.107635617850217560071374193435, −7.980395253428300847907467820821, −6.94646205341092811241512731591, −5.43088154739179142616086833158, −4.35669123879986398134206051716, −2.88269957781567645021396316359, −0.72213723308991920344142610595, 0.72213723308991920344142610595, 2.88269957781567645021396316359, 4.35669123879986398134206051716, 5.43088154739179142616086833158, 6.94646205341092811241512731591, 7.980395253428300847907467820821, 9.107635617850217560071374193435, 10.05956102446464702736683688807, 10.60038750242078630535181495454, 12.37396704661604279665517269154

Graph of the ZZ-function along the critical line