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Properties

Label 2-312-1.1-c3-0-3
Degree 22
Conductor 312312
Sign 11
Analytic cond. 18.408518.4085
Root an. cond. 4.290524.29052
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 18.8·5-s − 17.9·7-s + 9·9-s + 44.5·11-s + 13·13-s − 56.6·15-s + 34·17-s − 152.·19-s + 53.7·21-s + 107.·23-s + 231.·25-s − 27·27-s + 149.·29-s + 0.162·31-s − 133.·33-s − 338.·35-s − 256.·37-s − 39·39-s + 414.·41-s + 471.·43-s + 169.·45-s + 632.·47-s − 21.6·49-s − 102·51-s − 236.·53-s + 841.·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.68·5-s − 0.967·7-s + 0.333·9-s + 1.22·11-s + 0.277·13-s − 0.975·15-s + 0.485·17-s − 1.84·19-s + 0.558·21-s + 0.972·23-s + 1.85·25-s − 0.192·27-s + 0.958·29-s + 0.000943·31-s − 0.705·33-s − 1.63·35-s − 1.13·37-s − 0.160·39-s + 1.57·41-s + 1.67·43-s + 0.562·45-s + 1.96·47-s − 0.0630·49-s − 0.280·51-s − 0.612·53-s + 2.06·55-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: 11
Analytic conductor: 18.408518.4085
Root analytic conductor: 4.290524.29052
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 312, ( :3/2), 1)(2,\ 312,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.0258629882.025862988
L(12)L(\frac12) ≈\approx 2.0258629882.025862988
L(52)L(\frac{5}{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+3T 1 + 3T
13 1−13T 1 - 13T
good5 1−18.8T+125T2 1 - 18.8T + 125T^{2}
7 1+17.9T+343T2 1 + 17.9T + 343T^{2}
11 1−44.5T+1.33e3T2 1 - 44.5T + 1.33e3T^{2}
17 1−34T+4.91e3T2 1 - 34T + 4.91e3T^{2}
19 1+152.T+6.85e3T2 1 + 152.T + 6.85e3T^{2}
23 1−107.T+1.21e4T2 1 - 107.T + 1.21e4T^{2}
29 1−149.T+2.43e4T2 1 - 149.T + 2.43e4T^{2}
31 1−0.162T+2.97e4T2 1 - 0.162T + 2.97e4T^{2}
37 1+256.T+5.06e4T2 1 + 256.T + 5.06e4T^{2}
41 1−414.T+6.89e4T2 1 - 414.T + 6.89e4T^{2}
43 1−471.T+7.95e4T2 1 - 471.T + 7.95e4T^{2}
47 1−632.T+1.03e5T2 1 - 632.T + 1.03e5T^{2}
53 1+236.T+1.48e5T2 1 + 236.T + 1.48e5T^{2}
59 1+108.T+2.05e5T2 1 + 108.T + 2.05e5T^{2}
61 1−888.T+2.26e5T2 1 - 888.T + 2.26e5T^{2}
67 1−637.T+3.00e5T2 1 - 637.T + 3.00e5T^{2}
71 1+362.T+3.57e5T2 1 + 362.T + 3.57e5T^{2}
73 1−723.T+3.89e5T2 1 - 723.T + 3.89e5T^{2}
79 1+964.T+4.93e5T2 1 + 964.T + 4.93e5T^{2}
83 1+431.T+5.71e5T2 1 + 431.T + 5.71e5T^{2}
89 1−117.T+7.04e5T2 1 - 117.T + 7.04e5T^{2}
97 1+1.15e3T+9.12e5T2 1 + 1.15e3T + 9.12e5T^{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

−11.01788255615281080044484688719, −10.29021215663515932674085102953, −9.434161759233823153514538101846, −8.795359260173360243040948108278, −6.85588928254275163404796722296, −6.31925337851100174766700954760, −5.55812580269464539366589043322, −4.10833045461729011644367023953, −2.49479002275101306809856162620, −1.08246744018822829343940846184, 1.08246744018822829343940846184, 2.49479002275101306809856162620, 4.10833045461729011644367023953, 5.55812580269464539366589043322, 6.31925337851100174766700954760, 6.85588928254275163404796722296, 8.795359260173360243040948108278, 9.434161759233823153514538101846, 10.29021215663515932674085102953, 11.01788255615281080044484688719

Graph of the ZZ-function along the critical line