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Properties

Label 2-294-1.1-c3-0-0
Degree 22
Conductor 294294
Sign 11
Analytic cond. 17.346517.3465
Root an. cond. 4.164924.16492
Motivic weight 33
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 − 3·3-s + 4·4-s − 15·5-s + 6·6-s − 8·8-s + 9·9-s + 30·10-s − 9·11-s − 12·12-s − 88·13-s + 45·15-s + 16·16-s − 84·17-s − 18·18-s + 104·19-s − 60·20-s + 18·22-s − 84·23-s + 24·24-s + 100·25-s + 176·26-s − 27·27-s + 51·29-s − 90·30-s + 185·31-s − 32·32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.246·11-s − 0.288·12-s − 1.87·13-s + 0.774·15-s + 1/4·16-s − 1.19·17-s − 0.235·18-s + 1.25·19-s − 0.670·20-s + 0.174·22-s − 0.761·23-s + 0.204·24-s + 4/5·25-s + 1.32·26-s − 0.192·27-s + 0.326·29-s − 0.547·30-s + 1.07·31-s − 0.176·32-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    2⋅3⋅722 \cdot 3 \cdot 7^{2}
Sign: 11
Analytic conductor: 17.346517.3465
Root analytic conductor: 4.164924.16492
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 294, ( :3/2), 1)(2,\ 294,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.44036047480.4403604748
L(12)L(\frac12) ≈\approx 0.44036047480.4403604748
L(52)L(\frac{5}{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)
bad2 1+pT 1 + p T
3 1+pT 1 + p T
7 1 1
good5 1+3pT+p3T2 1 + 3 p T + p^{3} T^{2}
11 1+9T+p3T2 1 + 9 T + p^{3} T^{2}
13 1+88T+p3T2 1 + 88 T + p^{3} T^{2}
17 1+84T+p3T2 1 + 84 T + p^{3} T^{2}
19 1−104T+p3T2 1 - 104 T + p^{3} T^{2}
23 1+84T+p3T2 1 + 84 T + p^{3} T^{2}
29 1−51T+p3T2 1 - 51 T + p^{3} T^{2}
31 1−185T+p3T2 1 - 185 T + p^{3} T^{2}
37 1−44T+p3T2 1 - 44 T + p^{3} T^{2}
41 1+168T+p3T2 1 + 168 T + p^{3} T^{2}
43 1−326T+p3T2 1 - 326 T + p^{3} T^{2}
47 1+138T+p3T2 1 + 138 T + p^{3} T^{2}
53 1−639T+p3T2 1 - 639 T + p^{3} T^{2}
59 1−159T+p3T2 1 - 159 T + p^{3} T^{2}
61 1−722T+p3T2 1 - 722 T + p^{3} T^{2}
67 1+166T+p3T2 1 + 166 T + p^{3} T^{2}
71 1−1086T+p3T2 1 - 1086 T + p^{3} T^{2}
73 1−218T+p3T2 1 - 218 T + p^{3} T^{2}
79 1+583T+p3T2 1 + 583 T + p^{3} T^{2}
83 1+597T+p3T2 1 + 597 T + p^{3} T^{2}
89 1+1038T+p3T2 1 + 1038 T + p^{3} T^{2}
97 1+169T+p3T2 1 + 169 T + p^{3} 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

−11.56596781409439137732147752254, −10.36667715505423493958030194388, −9.586033762930345402568021951402, −8.311615210475905818810168784598, −7.49089621844068345015943072954, −6.80029112178917897706587837891, −5.23585305309818977553594612369, −4.15905706586965222923754467924, −2.55103126657328652305575389491, −0.50486537372711924433979769729, 0.50486537372711924433979769729, 2.55103126657328652305575389491, 4.15905706586965222923754467924, 5.23585305309818977553594612369, 6.80029112178917897706587837891, 7.49089621844068345015943072954, 8.311615210475905818810168784598, 9.586033762930345402568021951402, 10.36667715505423493958030194388, 11.56596781409439137732147752254

Graph of the ZZ-function along the critical line