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Properties

Label 2-6900-1.1-c1-0-19
Degree 22
Conductor 69006900
Sign 11
Analytic cond. 55.096755.0967
Root an. cond. 7.422727.42272
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  − 3-s + 3·7-s + 9-s − 2·11-s + 2·13-s + 7·17-s − 6·19-s − 3·21-s − 23-s − 27-s − 9·29-s + 9·31-s + 2·33-s + 7·37-s − 2·39-s + 5·41-s − 8·47-s + 2·49-s − 7·51-s + 11·53-s + 6·57-s + 9·59-s + 3·63-s + 3·67-s + 69-s + 3·71-s + 6·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.13·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 1.69·17-s − 1.37·19-s − 0.654·21-s − 0.208·23-s − 0.192·27-s − 1.67·29-s + 1.61·31-s + 0.348·33-s + 1.15·37-s − 0.320·39-s + 0.780·41-s − 1.16·47-s + 2/7·49-s − 0.980·51-s + 1.51·53-s + 0.794·57-s + 1.17·59-s + 0.377·63-s + 0.366·67-s + 0.120·69-s + 0.356·71-s + 0.702·73-s + ⋯

Functional equation

Λ(s)=(6900s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(6900s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 69006900    =    22⋅3⋅52⋅232^{2} \cdot 3 \cdot 5^{2} \cdot 23
Sign: 11
Analytic conductor: 55.096755.0967
Root analytic conductor: 7.422727.42272
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 6900, ( :1/2), 1)(2,\ 6900,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9185863911.918586391
L(12)L(\frac12) ≈\approx 1.9185863911.918586391
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 1
3 1+T 1 + T
5 1 1
23 1+T 1 + T
good7 1−3T+pT2 1 - 3 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1−7T+pT2 1 - 7 T + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
29 1+9T+pT2 1 + 9 T + p T^{2}
31 1−9T+pT2 1 - 9 T + p T^{2}
37 1−7T+pT2 1 - 7 T + p T^{2}
41 1−5T+pT2 1 - 5 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1−11T+pT2 1 - 11 T + p T^{2}
59 1−9T+pT2 1 - 9 T + p T^{2}
61 1+pT2 1 + p T^{2}
67 1−3T+pT2 1 - 3 T + p T^{2}
71 1−3T+pT2 1 - 3 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+5T+pT2 1 + 5 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1−10T+pT2 1 - 10 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

−8.069669155801640894146017525841, −7.36913090045138333376970374213, −6.44660273835631511784393340976, −5.73674719055418870802783672106, −5.24541142164515214520039056454, −4.41742680466752310626581711630, −3.77422385356999700348037806420, −2.61537486554548359682980367336, −1.69346935016057027327943982766, −0.76122223760427925475297547796, 0.76122223760427925475297547796, 1.69346935016057027327943982766, 2.61537486554548359682980367336, 3.77422385356999700348037806420, 4.41742680466752310626581711630, 5.24541142164515214520039056454, 5.73674719055418870802783672106, 6.44660273835631511784393340976, 7.36913090045138333376970374213, 8.069669155801640894146017525841

Graph of the ZZ-function along the critical line