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Properties

Label 2-138-1.1-c1-0-1
Degree 22
Conductor 138138
Sign 11
Analytic cond. 1.101931.10193
Root an. cond. 1.049731.04973
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 − 12-s − 2·13-s − 2·15-s + 16-s + 2·17-s + 18-s − 8·19-s + 2·20-s − 23-s − 24-s − 25-s − 2·26-s − 27-s − 2·29-s − 2·30-s − 8·31-s + 32-s + 2·34-s + 36-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 − 0.288·12-s − 0.554·13-s − 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 1.83·19-s + 0.447·20-s − 0.208·23-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 − 1.43·31-s + 0.176·32-s + 0.342·34-s + 1/6·36-s + ⋯

Functional equation

Λ(s)=(138s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(138s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 138138    =    2⋅3⋅232 \cdot 3 \cdot 23
Sign: 11
Analytic conductor: 1.101931.10193
Root analytic conductor: 1.049731.04973
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 138, ( :1/2), 1)(2,\ 138,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4877307681.487730768
L(12)L(\frac12) ≈\approx 1.4877307681.487730768
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)Isogeny Class over Fp\mathbf{F}_p
bad2 1−T 1 - T
3 1+T 1 + T
23 1+T 1 + T
good5 1−2T+pT2 1 - 2 T + p T^{2} 1.5.ac
7 1+pT2 1 + p T^{2} 1.7.a
11 1+pT2 1 + p T^{2} 1.11.a
13 1+2T+pT2 1 + 2 T + p T^{2} 1.13.c
17 1−2T+pT2 1 - 2 T + p T^{2} 1.17.ac
19 1+8T+pT2 1 + 8 T + p T^{2} 1.19.i
29 1+2T+pT2 1 + 2 T + p T^{2} 1.29.c
31 1+8T+pT2 1 + 8 T + p T^{2} 1.31.i
37 1−2T+pT2 1 - 2 T + p T^{2} 1.37.ac
41 1−10T+pT2 1 - 10 T + p T^{2} 1.41.ak
43 1−8T+pT2 1 - 8 T + p T^{2} 1.43.ai
47 1−8T+pT2 1 - 8 T + p T^{2} 1.47.ai
53 1−2T+pT2 1 - 2 T + p T^{2} 1.53.ac
59 1+4T+pT2 1 + 4 T + p T^{2} 1.59.e
61 1−2T+pT2 1 - 2 T + p T^{2} 1.61.ac
67 1−8T+pT2 1 - 8 T + p T^{2} 1.67.ai
71 1+pT2 1 + p T^{2} 1.71.a
73 1+6T+pT2 1 + 6 T + p T^{2} 1.73.g
79 1−8T+pT2 1 - 8 T + p T^{2} 1.79.ai
83 1+16T+pT2 1 + 16 T + p T^{2} 1.83.q
89 1−18T+pT2 1 - 18 T + p T^{2} 1.89.as
97 1−10T+pT2 1 - 10 T + p T^{2} 1.97.ak
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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

−12.99091917601195955204136966309, −12.47239235999888074523687731605, −11.18195471039845683279157830959, −10.34314478687064358780223455819, −9.229114075576505473092106150497, −7.56963733345572718426271943739, −6.29398123690432140072875572952, −5.50969858567909471909928316815, −4.17219923557025010086906698339, −2.18807119288166115407408001809, 2.18807119288166115407408001809, 4.17219923557025010086906698339, 5.50969858567909471909928316815, 6.29398123690432140072875572952, 7.56963733345572718426271943739, 9.229114075576505473092106150497, 10.34314478687064358780223455819, 11.18195471039845683279157830959, 12.47239235999888074523687731605, 12.99091917601195955204136966309

Graph of the ZZ-function along the critical line