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Properties

Label 2-234-1.1-c1-0-4
Degree 22
Conductor 234234
Sign −1-1
Analytic cond. 1.868491.86849
Root an. cond. 1.366931.36693
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 + 4-s − 2·5-s − 2·7-s − 8-s + 2·10-s − 4·11-s − 13-s + 2·14-s + 16-s − 6·19-s − 2·20-s + 4·22-s + 4·23-s − 25-s + 26-s − 2·28-s − 8·29-s − 2·31-s − 32-s + 4·35-s + 6·37-s + 6·38-s + 2·40-s + 6·41-s − 8·43-s − 4·44-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.755·7-s − 0.353·8-s + 0.632·10-s − 1.20·11-s − 0.277·13-s + 0.534·14-s + 1/4·16-s − 1.37·19-s − 0.447·20-s + 0.852·22-s + 0.834·23-s − 1/5·25-s + 0.196·26-s − 0.377·28-s − 1.48·29-s − 0.359·31-s − 0.176·32-s + 0.676·35-s + 0.986·37-s + 0.973·38-s + 0.316·40-s + 0.937·41-s − 1.21·43-s − 0.603·44-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: −1-1
Analytic conductor: 1.868491.86849
Root analytic conductor: 1.366931.36693
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 234, ( :1/2), −1)(2,\ 234,\ (\ :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 1
13 1+T 1 + T
good5 1+2T+pT2 1 + 2 T + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+8T+pT2 1 + 8 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 1−6T+pT2 1 - 6 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1−12T+pT2 1 - 12 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1+16T+pT2 1 + 16 T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+6T+pT2 1 + 6 T + 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

−11.48471457324402872742962494510, −10.70725444808913514749917450090, −9.773699703719295926726447008045, −8.695739660347841822203445826112, −7.76516545555292037952264969735, −6.93707785293490008685942511031, −5.59584593532556316161391417712, −3.99394475360161838020150844401, −2.57523735650192716532994366231, 0, 2.57523735650192716532994366231, 3.99394475360161838020150844401, 5.59584593532556316161391417712, 6.93707785293490008685942511031, 7.76516545555292037952264969735, 8.695739660347841822203445826112, 9.773699703719295926726447008045, 10.70725444808913514749917450090, 11.48471457324402872742962494510

Graph of the ZZ-function along the critical line