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Properties

Label 2-175-1.1-c1-0-7
Degree 22
Conductor 175175
Sign 11
Analytic cond. 1.397381.39738
Root an. cond. 1.182101.18210
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 2·4-s + 2·6-s − 7-s − 2·9-s − 3·11-s + 2·12-s + 13-s − 2·14-s − 4·16-s + 7·17-s − 4·18-s − 21-s − 6·22-s + 6·23-s + 2·26-s − 5·27-s − 2·28-s − 5·29-s + 2·31-s − 8·32-s − 3·33-s + 14·34-s − 4·36-s + 2·37-s + 39-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s − 0.377·7-s − 2/3·9-s − 0.904·11-s + 0.577·12-s + 0.277·13-s − 0.534·14-s − 16-s + 1.69·17-s − 0.942·18-s − 0.218·21-s − 1.27·22-s + 1.25·23-s + 0.392·26-s − 0.962·27-s − 0.377·28-s − 0.928·29-s + 0.359·31-s − 1.41·32-s − 0.522·33-s + 2.40·34-s − 2/3·36-s + 0.328·37-s + 0.160·39-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 11
Analytic conductor: 1.397381.39738
Root analytic conductor: 1.182101.18210
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 175, ( :1/2), 1)(2,\ 175,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.3474425342.347442534
L(12)L(\frac12) ≈\approx 2.3474425342.347442534
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)
bad5 1 1
7 1+T 1 + T
good2 1−pT+pT2 1 - p T + p T^{2}
3 1−T+pT2 1 - T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
17 1−7T+pT2 1 - 7 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1+5T+pT2 1 + 5 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+3T+pT2 1 + 3 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−10T+pT2 1 - 10 T + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1−2T+pT2 1 - 2 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1+5T+pT2 1 + 5 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1−7T+pT2 1 - 7 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

−13.01254965419346541735155009772, −12.04804768145326897305333226930, −11.05181111943764343536756650509, −9.734749354117359778149997929022, −8.569343753087953468815992872802, −7.38451703018051703490900915946, −5.93898740434763322465740513982, −5.14936161735373762545181523253, −3.57752582126466520607951640819, −2.76472422088648867371864611007, 2.76472422088648867371864611007, 3.57752582126466520607951640819, 5.14936161735373762545181523253, 5.93898740434763322465740513982, 7.38451703018051703490900915946, 8.569343753087953468815992872802, 9.734749354117359778149997929022, 11.05181111943764343536756650509, 12.04804768145326897305333226930, 13.01254965419346541735155009772

Graph of the ZZ-function along the critical line