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Properties

Label 2-3234-1.1-c1-0-24
Degree 22
Conductor 32343234
Sign 11
Analytic cond. 25.823625.8236
Root an. cond. 5.081695.08169
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-s + 4-s + 2·5-s − 6-s − 8-s + 9-s − 2·10-s + 11-s + 12-s − 2·13-s + 2·15-s + 16-s + 6·17-s − 18-s + 4·19-s + 2·20-s − 22-s − 4·23-s − 24-s − 25-s + 2·26-s + 27-s + 2·29-s − 2·30-s + 4·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.288·12-s − 0.554·13-s + 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s − 0.213·22-s − 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.371·29-s − 0.365·30-s + 0.718·31-s − 0.176·32-s + ⋯

Functional equation

Λ(s)=(3234s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3234s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32343234    =    2⋅3⋅72⋅112 \cdot 3 \cdot 7^{2} \cdot 11
Sign: 11
Analytic conductor: 25.823625.8236
Root analytic conductor: 5.081695.08169
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3234, ( :1/2), 1)(2,\ 3234,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1546360012.154636001
L(12)L(\frac12) ≈\approx 2.1546360012.154636001
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−T 1 - T
7 1 1
11 1−T 1 - T
good5 1−2T+pT2 1 - 2 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1+14T+pT2 1 + 14 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−14T+pT2 1 - 14 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

−8.736150936012440857648978735320, −7.84439006039971069748622971041, −7.49024027339887201227457052610, −6.43679449975526542026435018487, −5.79323544635467042022952173625, −4.93054069578340184917711747999, −3.71446826599373971796272627437, −2.83711647406054313257956134718, −1.96943863395594925899870544120, −1.00454239585262764496288710387, 1.00454239585262764496288710387, 1.96943863395594925899870544120, 2.83711647406054313257956134718, 3.71446826599373971796272627437, 4.93054069578340184917711747999, 5.79323544635467042022952173625, 6.43679449975526542026435018487, 7.49024027339887201227457052610, 7.84439006039971069748622971041, 8.736150936012440857648978735320

Graph of the ZZ-function along the critical line