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Properties

Label 4-1110e2-1.1-c1e2-0-3
Degree 44
Conductor 12321001232100
Sign 11
Analytic cond. 78.559778.5597
Root an. cond. 2.977142.97714
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 5-s + 6-s − 2·7-s − 8-s + 10-s − 10·11-s − 5·13-s − 2·14-s + 15-s − 16-s + 5·17-s − 2·19-s − 2·21-s − 10·22-s − 24-s − 5·26-s − 27-s − 18·29-s + 30-s + 8·31-s − 10·33-s + 5·34-s − 2·35-s + 37-s − 2·38-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 0.316·10-s − 3.01·11-s − 1.38·13-s − 0.534·14-s + 0.258·15-s − 1/4·16-s + 1.21·17-s − 0.458·19-s − 0.436·21-s − 2.13·22-s − 0.204·24-s − 0.980·26-s − 0.192·27-s − 3.34·29-s + 0.182·30-s + 1.43·31-s − 1.74·33-s + 0.857·34-s − 0.338·35-s + 0.164·37-s − 0.324·38-s + ⋯

Functional equation

Λ(s)=(1232100s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1232100s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 12321001232100    =    22⋅32⋅52⋅3722^{2} \cdot 3^{2} \cdot 5^{2} \cdot 37^{2}
Sign: 11
Analytic conductor: 78.559778.5597
Root analytic conductor: 2.977142.97714
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1232100, ( :1/2,1/2), 1)(4,\ 1232100,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1321811111.132181111
L(12)L(\frac12) ≈\approx 1.1321811111.132181111
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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C2C_2 1−T+T2 1 - T + T^{2}
3C2C_2 1−T+T2 1 - T + T^{2}
5C2C_2 1−T+T2 1 - T + T^{2}
37C2C_2 1−T+pT2 1 - T + p T^{2}
good7C22C_2^2 1+2T−3T2+2pT3+p2T4 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4}
11C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
13C2C_2 (1−2T+pT2)(1+7T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} )
17C22C_2^2 1−5T+8T2−5pT3+p2T4 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4}
19C22C_2^2 1+2T−15T2+2pT3+p2T4 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
31C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
41C22C_2^2 1+6T−5T2+6pT3+p2T4 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4}
43C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
47C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
53C22C_2^2 1+6T−17T2+6pT3+p2T4 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4}
59C22C_2^2 1+3T−50T2+3pT3+p2T4 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
67C22C_2^2 1−9T+14T2−9pT3+p2T4 1 - 9 T + 14 T^{2} - 9 p T^{3} + p^{2} T^{4}
71C22C_2^2 1−12T+73T2−12pT3+p2T4 1 - 12 T + 73 T^{2} - 12 p T^{3} + p^{2} T^{4}
73C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
79C22C_2^2 1+16T+177T2+16pT3+p2T4 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4}
83C22C_2^2 1−2T−79T2−2pT3+p2T4 1 - 2 T - 79 T^{2} - 2 p T^{3} + p^{2} T^{4}
89C22C_2^2 1−14T+107T2−14pT3+p2T4 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4}
97C2C_2 (1+18T+pT2)2 ( 1 + 18 T + p T^{2} )^{2}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.895229605154386705753371611054, −9.830617483174062762974913241447, −9.258885623341698062529608626560, −8.952677386355150180209916458371, −8.279524988625482730407823827717, −7.79388202743721958840015857397, −7.67797784157869321364174352747, −7.27202117645734074763766748881, −6.79331068334586798801399775171, −5.92412104781530460098390377473, −5.67999674593804376707799382122, −5.25889641366113415656529714713, −5.22187997594200934365590020570, −4.17533733727464206227780713276, −4.07700134980005773011811315758, −3.01787900420208888078971942426, −2.91216122600353537452807139657, −2.46533774132156703503218947021, −1.90660766179344916812031405289, −0.36968190955454456474643329672, 0.36968190955454456474643329672, 1.90660766179344916812031405289, 2.46533774132156703503218947021, 2.91216122600353537452807139657, 3.01787900420208888078971942426, 4.07700134980005773011811315758, 4.17533733727464206227780713276, 5.22187997594200934365590020570, 5.25889641366113415656529714713, 5.67999674593804376707799382122, 5.92412104781530460098390377473, 6.79331068334586798801399775171, 7.27202117645734074763766748881, 7.67797784157869321364174352747, 7.79388202743721958840015857397, 8.279524988625482730407823827717, 8.952677386355150180209916458371, 9.258885623341698062529608626560, 9.830617483174062762974913241447, 9.895229605154386705753371611054

Graph of the ZZ-function along the critical line