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Properties

Label 2-1078-1.1-c1-0-29
Degree 22
Conductor 10781078
Sign 11
Analytic cond. 8.607878.60787
Root an. cond. 2.933912.93391
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-s + 3·3-s + 4-s + 2·5-s + 3·6-s + 8-s + 6·9-s + 2·10-s − 11-s + 3·12-s − 7·13-s + 6·15-s + 16-s + 2·17-s + 6·18-s + 2·20-s − 22-s − 8·23-s + 3·24-s − 25-s − 7·26-s + 9·27-s − 5·29-s + 6·30-s + 4·31-s + 32-s − 3·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 1/2·4-s + 0.894·5-s + 1.22·6-s + 0.353·8-s + 2·9-s + 0.632·10-s − 0.301·11-s + 0.866·12-s − 1.94·13-s + 1.54·15-s + 1/4·16-s + 0.485·17-s + 1.41·18-s + 0.447·20-s − 0.213·22-s − 1.66·23-s + 0.612·24-s − 1/5·25-s − 1.37·26-s + 1.73·27-s − 0.928·29-s + 1.09·30-s + 0.718·31-s + 0.176·32-s − 0.522·33-s + ⋯

Functional equation

Λ(s)=(1078s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1078s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10781078    =    2⋅72⋅112 \cdot 7^{2} \cdot 11
Sign: 11
Analytic conductor: 8.607878.60787
Root analytic conductor: 2.933912.93391
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1078, ( :1/2), 1)(2,\ 1078,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.6856528294.685652829
L(12)L(\frac12) ≈\approx 4.6856528294.685652829
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
7 1 1
11 1+T 1 + T
good3 1−pT+pT2 1 - p T + p T^{2}
5 1−2T+pT2 1 - 2 T + p T^{2}
13 1+7T+pT2 1 + 7 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1+5T+pT2 1 + 5 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1−4T+pT2 1 - 4 T + p T^{2}
41 1−4T+pT2 1 - 4 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1−2T+pT2 1 - 2 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1−3T+pT2 1 - 3 T + p T^{2}
61 1−T+pT2 1 - T + p T^{2}
67 1−9T+pT2 1 - 9 T + p T^{2}
71 1+2T+pT2 1 + 2 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1−9T+pT2 1 - 9 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1−6T+pT2 1 - 6 T + 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

−9.856353077622187691368346097215, −9.232984660708510751075852428852, −7.949688248994453332985143052492, −7.66334045549742507331716148152, −6.56153010353145194477530352450, −5.46355028016127947092103451217, −4.50301199161709359138099430647, −3.48832092996576829359255750084, −2.43990333774251370043130333163, −2.00011124513055881492662257488, 2.00011124513055881492662257488, 2.43990333774251370043130333163, 3.48832092996576829359255750084, 4.50301199161709359138099430647, 5.46355028016127947092103451217, 6.56153010353145194477530352450, 7.66334045549742507331716148152, 7.949688248994453332985143052492, 9.232984660708510751075852428852, 9.856353077622187691368346097215

Graph of the ZZ-function along the critical line