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Properties

Label 2-103428-1.1-c1-0-17
Degree 22
Conductor 103428103428
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 4·7-s − 11-s + 17-s + 3·19-s + 8·23-s − 25-s + 5·29-s + 2·31-s + 8·35-s + 6·37-s + 3·41-s − 11·43-s + 2·47-s + 9·49-s − 12·53-s + 2·55-s − 2·59-s − 8·61-s − 7·67-s + 8·71-s + 4·73-s + 4·77-s − 10·79-s + 6·83-s − 2·85-s + 6·89-s + ⋯
L(s)  = 1  − 0.894·5-s − 1.51·7-s − 0.301·11-s + 0.242·17-s + 0.688·19-s + 1.66·23-s − 1/5·25-s + 0.928·29-s + 0.359·31-s + 1.35·35-s + 0.986·37-s + 0.468·41-s − 1.67·43-s + 0.291·47-s + 9/7·49-s − 1.64·53-s + 0.269·55-s − 0.260·59-s − 1.02·61-s − 0.855·67-s + 0.949·71-s + 0.468·73-s + 0.455·77-s − 1.12·79-s + 0.658·83-s − 0.216·85-s + 0.635·89-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: −1-1
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: 11
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+2T+pT2 1 + 2 T + p T^{2}
7 1+4T+pT2 1 + 4 T + p T^{2}
11 1+T+pT2 1 + T + p T^{2}
19 1−3T+pT2 1 - 3 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1−5T+pT2 1 - 5 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1−3T+pT2 1 - 3 T + p T^{2}
43 1+11T+pT2 1 + 11 T + p T^{2}
47 1−2T+pT2 1 - 2 T + p T^{2}
53 1+12T+pT2 1 + 12 T + p T^{2}
59 1+2T+pT2 1 + 2 T + p T^{2}
61 1+8T+pT2 1 + 8 T + p T^{2}
67 1+7T+pT2 1 + 7 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1+10T+pT2 1 + 10 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+2T+pT2 1 + 2 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

−13.81963982104613, −13.45921332359443, −12.77400924929024, −12.72300258779229, −11.94840200706077, −11.65882115382854, −11.02660799016772, −10.55744695777945, −9.939371885131566, −9.568584797503604, −9.093876466679915, −8.504515968729625, −7.891987678715868, −7.462137643895979, −6.978743798093645, −6.286717138949322, −6.127662541813103, −5.091742479871928, −4.836566864946190, −4.017029460724087, −3.451777023526650, −2.993710645391235, −2.641682179684229, −1.448119621076965, −0.7153927441027272, 0, 0.7153927441027272, 1.448119621076965, 2.641682179684229, 2.993710645391235, 3.451777023526650, 4.017029460724087, 4.836566864946190, 5.091742479871928, 6.127662541813103, 6.286717138949322, 6.978743798093645, 7.462137643895979, 7.891987678715868, 8.504515968729625, 9.093876466679915, 9.568584797503604, 9.939371885131566, 10.55744695777945, 11.02660799016772, 11.65882115382854, 11.94840200706077, 12.72300258779229, 12.77400924929024, 13.45921332359443, 13.81963982104613

Graph of the ZZ-function along the critical line