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Properties

Label 4-3240e2-1.1-c1e2-0-31
Degree 44
Conductor 1049760010497600
Sign 11
Analytic cond. 669.336669.336
Root an. cond. 5.086405.08640
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 2·7-s + 6·13-s + 14·17-s + 14·19-s − 7·23-s − 6·29-s − 3·31-s + 2·35-s − 12·37-s − 4·41-s − 8·43-s + 4·47-s + 7·49-s − 10·53-s − 6·59-s + 3·61-s + 6·65-s + 10·67-s + 24·71-s + 32·73-s − 79-s − 9·83-s + 14·85-s − 8·89-s + 12·91-s + 14·95-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.755·7-s + 1.66·13-s + 3.39·17-s + 3.21·19-s − 1.45·23-s − 1.11·29-s − 0.538·31-s + 0.338·35-s − 1.97·37-s − 0.624·41-s − 1.21·43-s + 0.583·47-s + 49-s − 1.37·53-s − 0.781·59-s + 0.384·61-s + 0.744·65-s + 1.22·67-s + 2.84·71-s + 3.74·73-s − 0.112·79-s − 0.987·83-s + 1.51·85-s − 0.847·89-s + 1.25·91-s + 1.43·95-s + ⋯

Functional equation

Λ(s)=(10497600s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(10497600s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 1049760010497600    =    26⋅38⋅522^{6} \cdot 3^{8} \cdot 5^{2}
Sign: 11
Analytic conductor: 669.336669.336
Root analytic conductor: 5.086405.08640
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 10497600, ( :1/2,1/2), 1)(4,\ 10497600,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 5.3317387955.331738795
L(12)L(\frac12) ≈\approx 5.3317387955.331738795
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)
bad2 1 1
3 1 1
5C2C_2 1−T+T2 1 - T + T^{2}
good7C22C_2^2 1−2T−3T2−2pT3+p2T4 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4}
11C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
13C22C_2^2 1−6T+23T2−6pT3+p2T4 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4}
17C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
19C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
23C22C_2^2 1+7T+26T2+7pT3+p2T4 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4}
29C22C_2^2 1+6T+7T2+6pT3+p2T4 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4}
31C22C_2^2 1+3T−22T2+3pT3+p2T4 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4}
37C2C_2 (1+6T+pT2)2 ( 1 + 6 T + p T^{2} )^{2}
41C22C_2^2 1+4T−25T2+4pT3+p2T4 1 + 4 T - 25 T^{2} + 4 p T^{3} + p^{2} T^{4}
43C2C_2 (1−5T+pT2)(1+13T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} )
47C22C_2^2 1−4T−31T2−4pT3+p2T4 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4}
53C2C_2 (1+5T+pT2)2 ( 1 + 5 T + p T^{2} )^{2}
59C22C_2^2 1+6T−23T2+6pT3+p2T4 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−3T−52T2−3pT3+p2T4 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4}
67C22C_2^2 1−10T+33T2−10pT3+p2T4 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4}
71C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
73C2C_2 (1−16T+pT2)2 ( 1 - 16 T + p T^{2} )^{2}
79C22C_2^2 1+T−78T2+pT3+p2T4 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4}
83C22C_2^2 1+9T−2T2+9pT3+p2T4 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4}
89C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
97C22C_2^2 1−16T+159T2−16pT3+p2T4 1 - 16 T + 159 T^{2} - 16 p T^{3} + p^{2} T^{4}
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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

−8.645056597312782358062323748869, −8.451659412786848327973492643791, −7.938982820141644406833992380808, −7.889871179447226501993421457074, −7.37468764244695724052393142993, −7.18217105161627753759527882209, −6.55422510419272755035961748222, −6.06027471198198289740930481213, −5.65771761174981089982074949307, −5.40800091009319753371250765394, −5.22574979274172579226159946548, −4.90041531420908580183324751915, −3.84390215677984779011606653403, −3.62419632611300659058531839053, −3.30909039291830617980713266152, −3.21314298034999413343180849806, −1.95005600693802729060898423601, −1.83557380284143849680891655418, −1.06672505350067614836791233750, −0.903346765991348911470250951074, 0.903346765991348911470250951074, 1.06672505350067614836791233750, 1.83557380284143849680891655418, 1.95005600693802729060898423601, 3.21314298034999413343180849806, 3.30909039291830617980713266152, 3.62419632611300659058531839053, 3.84390215677984779011606653403, 4.90041531420908580183324751915, 5.22574979274172579226159946548, 5.40800091009319753371250765394, 5.65771761174981089982074949307, 6.06027471198198289740930481213, 6.55422510419272755035961748222, 7.18217105161627753759527882209, 7.37468764244695724052393142993, 7.889871179447226501993421457074, 7.938982820141644406833992380808, 8.451659412786848327973492643791, 8.645056597312782358062323748869

Graph of the ZZ-function along the critical line