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Properties

Label 2-5610-1.1-c1-0-107
Degree 22
Conductor 56105610
Sign −1-1
Analytic cond. 44.796044.7960
Root an. cond. 6.692986.69298
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-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 11-s + 12-s − 6·13-s − 15-s + 16-s + 17-s + 18-s − 20-s + 22-s − 8·23-s + 24-s + 25-s − 6·26-s + 27-s − 6·29-s − 30-s − 4·31-s + 32-s + 33-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 1.66·13-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.223·20-s + 0.213·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s − 1.11·29-s − 0.182·30-s − 0.718·31-s + 0.176·32-s + 0.174·33-s + ⋯

Functional equation

Λ(s)=(5610s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(5610s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 56105610    =    2⋅3⋅5⋅11⋅172 \cdot 3 \cdot 5 \cdot 11 \cdot 17
Sign: −1-1
Analytic conductor: 44.796044.7960
Root analytic conductor: 6.692986.69298
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 5610, ( :1/2), −1)(2,\ 5610,\ (\ :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)
bad2 1−T 1 - T
3 1−T 1 - T
5 1+T 1 + T
11 1−T 1 - T
17 1−T 1 - T
good7 1+pT2 1 + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+4T+pT2 1 + 4 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+8T+pT2 1 + 8 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−2T+pT2 1 - 2 T + p T^{2}
97 1+6T+pT2 1 + 6 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

−7.84252706042761906997047180712, −7.09356023726554257760712127854, −6.41917811141756616347522529543, −5.45960223715275273534634780220, −4.79172099955020537806345076640, −4.00310702636647721065480037791, −3.39860484802407122485507683606, −2.44587792994532802113306372656, −1.72243927296666951440031206183, 0, 1.72243927296666951440031206183, 2.44587792994532802113306372656, 3.39860484802407122485507683606, 4.00310702636647721065480037791, 4.79172099955020537806345076640, 5.45960223715275273534634780220, 6.41917811141756616347522529543, 7.09356023726554257760712127854, 7.84252706042761906997047180712

Graph of the ZZ-function along the critical line