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Properties

Label 2-103428-1.1-c1-0-30
Degree 22
Conductor 103428103428
Sign 11
Analytic cond. 825.876825.876
Root an. cond. 28.738028.7380
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 22

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·11-s + 17-s − 2·19-s − 8·23-s − 5·25-s − 6·29-s + 4·31-s − 8·41-s − 8·43-s − 2·47-s − 7·49-s − 10·53-s − 6·59-s + 6·61-s + 14·67-s + 4·73-s − 8·79-s − 6·83-s + 14·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.20·11-s + 0.242·17-s − 0.458·19-s − 1.66·23-s − 25-s − 1.11·29-s + 0.718·31-s − 1.24·41-s − 1.21·43-s − 0.291·47-s − 49-s − 1.37·53-s − 0.781·59-s + 0.768·61-s + 1.71·67-s + 0.468·73-s − 0.900·79-s − 0.658·83-s + 1.48·89-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

Λ(s)=(103428s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 103428 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(103428s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 103428 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 103428103428    =    22⋅32⋅132⋅172^{2} \cdot 3^{2} \cdot 13^{2} \cdot 17
Sign: 11
Analytic conductor: 825.876825.876
Root analytic conductor: 28.738028.7380
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 22
Selberg data: (2, 103428, ( :1/2), 1)(2,\ 103428,\ (\ :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 1
3 1 1
13 1 1
17 1−T 1 - T
good5 1+pT2 1 + p T^{2}
7 1+pT2 1 + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
19 1+2T+pT2 1 + 2 T + 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+pT2 1 + p T^{2}
41 1+8T+pT2 1 + 8 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+10T+pT2 1 + 10 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1−14T+pT2 1 - 14 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1−14T+pT2 1 - 14 T + p T^{2}
97 1+8T+pT2 1 + 8 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

−14.14036941826513, −13.71970868637730, −13.19485105819244, −12.81180566553131, −12.31594436748843, −11.67340361099245, −11.37634541891120, −10.77112316494211, −10.12552900937330, −9.889496271713330, −9.483697895450507, −8.492614530672672, −8.278822631416714, −7.787447998892146, −7.335430261215389, −6.527369270493040, −6.149553794555361, −5.570541149369850, −4.980364700963020, −4.566402252990148, −3.627236460884990, −3.465844167908392, −2.465436294771985, −2.043128339928795, −1.382634312991862, 0, 0, 1.382634312991862, 2.043128339928795, 2.465436294771985, 3.465844167908392, 3.627236460884990, 4.566402252990148, 4.980364700963020, 5.570541149369850, 6.149553794555361, 6.527369270493040, 7.335430261215389, 7.787447998892146, 8.278822631416714, 8.492614530672672, 9.483697895450507, 9.889496271713330, 10.12552900937330, 10.77112316494211, 11.37634541891120, 11.67340361099245, 12.31594436748843, 12.81180566553131, 13.19485105819244, 13.71970868637730, 14.14036941826513

Graph of the ZZ-function along the critical line