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Properties

Label 2-2175-1.1-c1-0-84
Degree 22
Conductor 21752175
Sign −1-1
Analytic cond. 17.367417.3674
Root an. cond. 4.167424.16742
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3-s + 2·4-s − 2·6-s + 2·7-s + 9-s − 3·11-s − 2·12-s − 4·13-s + 4·14-s − 4·16-s − 8·17-s + 2·18-s − 2·21-s − 6·22-s + 23-s − 8·26-s − 27-s + 4·28-s + 29-s − 8·31-s − 8·32-s + 3·33-s − 16·34-s + 2·36-s + 7·37-s + 4·39-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s + 0.755·7-s + 1/3·9-s − 0.904·11-s − 0.577·12-s − 1.10·13-s + 1.06·14-s − 16-s − 1.94·17-s + 0.471·18-s − 0.436·21-s − 1.27·22-s + 0.208·23-s − 1.56·26-s − 0.192·27-s + 0.755·28-s + 0.185·29-s − 1.43·31-s − 1.41·32-s + 0.522·33-s − 2.74·34-s + 1/3·36-s + 1.15·37-s + 0.640·39-s + ⋯

Functional equation

Λ(s)=(2175s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2175s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21752175    =    3⋅52⋅293 \cdot 5^{2} \cdot 29
Sign: −1-1
Analytic conductor: 17.367417.3674
Root analytic conductor: 4.167424.16742
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2175, ( :1/2), −1)(2,\ 2175,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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)
bad3 1+T 1 + T
5 1 1
29 1−T 1 - T
good2 1−pT+pT2 1 - p T + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1+8T+pT2 1 + 8 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1−T+pT2 1 - T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1−7T+pT2 1 - 7 T + p T^{2}
41 1−7T+pT2 1 - 7 T + p T^{2}
43 1+9T+pT2 1 + 9 T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1+9T+pT2 1 + 9 T + p T^{2}
59 1−10T+pT2 1 - 10 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1−T+pT2 1 - T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1+9T+pT2 1 + 9 T + p T^{2}
89 1−10T+pT2 1 - 10 T + p T^{2}
97 1+13T+pT2 1 + 13 T + p T^{2}
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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

−8.663003477671988172128411423907, −7.57267794450033413725994868187, −6.94741208393912926694336652961, −6.03449239221672116236167058426, −5.28269572114788096995193818073, −4.68654456311297527430440074459, −4.15797257967427379453563556870, −2.79373927165665359734182776318, −2.04532514368342061778964974779, 0, 2.04532514368342061778964974779, 2.79373927165665359734182776318, 4.15797257967427379453563556870, 4.68654456311297527430440074459, 5.28269572114788096995193818073, 6.03449239221672116236167058426, 6.94741208393912926694336652961, 7.57267794450033413725994868187, 8.663003477671988172128411423907

Graph of the ZZ-function along the critical line