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Properties

Label 4-3591-1.1-c1e2-0-0
Degree 44
Conductor 35913591
Sign 11
Analytic cond. 0.2289650.228965
Root an. cond. 0.6917390.691739
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  − 3-s + 4-s + 7-s + 9-s − 12-s − 4·13-s − 3·16-s + 3·19-s − 21-s + 2·25-s − 27-s + 28-s + 36-s − 12·37-s + 4·39-s − 8·43-s + 3·48-s − 6·49-s − 4·52-s − 3·57-s + 20·61-s + 63-s − 7·64-s − 8·67-s + 4·73-s − 2·75-s + 3·76-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s + 0.377·7-s + 1/3·9-s − 0.288·12-s − 1.10·13-s − 3/4·16-s + 0.688·19-s − 0.218·21-s + 2/5·25-s − 0.192·27-s + 0.188·28-s + 1/6·36-s − 1.97·37-s + 0.640·39-s − 1.21·43-s + 0.433·48-s − 6/7·49-s − 0.554·52-s − 0.397·57-s + 2.56·61-s + 0.125·63-s − 7/8·64-s − 0.977·67-s + 0.468·73-s − 0.230·75-s + 0.344·76-s + ⋯

Functional equation

Λ(s)=(3591s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3591 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3591s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3591 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 35913591    =    33⋅7⋅193^{3} \cdot 7 \cdot 19
Sign: 11
Analytic conductor: 0.2289650.228965
Root analytic conductor: 0.6917390.691739
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 3591, ( :1/2,1/2), 1)(4,\ 3591,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.71753290720.7175329072
L(12)L(\frac12) ≈\approx 0.71753290720.7175329072
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)
bad3C1C_1 1+T 1 + T
7C1C_1×\timesC2C_2 (1−T)(1+pT2) ( 1 - T )( 1 + p T^{2} )
19C1C_1×\timesC2C_2 (1+T)(1−4T+pT2) ( 1 + T )( 1 - 4 T + p T^{2} )
good2C22C_2^2 1−T2+p2T4 1 - T^{2} + p^{2} T^{4}
5C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
11C22C_2^2 1−2T2+p2T4 1 - 2 T^{2} + p^{2} T^{4}
13C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
17C22C_2^2 1+22T2+p2T4 1 + 22 T^{2} + p^{2} T^{4}
23C22C_2^2 1−10T2+p2T4 1 - 10 T^{2} + p^{2} T^{4}
29C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2×\timesC2C_2 (1+2T+pT2)(1+10T+pT2) ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} )
41C2C_2 (1−10T+pT2)(1+10T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )
43C2C_2×\timesC2C_2 (1−4T+pT2)(1+12T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} )
47C22C_2^2 1+14T2+p2T4 1 + 14 T^{2} + p^{2} T^{4}
53C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
59C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
61C2C_2×\timesC2C_2 (1−14T+pT2)(1−6T+pT2) ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} )
67C2C_2×\timesC2C_2 (1−4T+pT2)(1+12T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} )
71C22C_2^2 1−34T2+p2T4 1 - 34 T^{2} + p^{2} T^{4}
73C2C_2×\timesC2C_2 (1−10T+pT2)(1+6T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} )
79C2C_2×\timesC2C_2 (1−16T+pT2)(1+8T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C22C_2^2 1−106T2+p2T4 1 - 106 T^{2} + p^{2} T^{4}
89C22C_2^2 1−82T2+p2T4 1 - 82 T^{2} + p^{2} T^{4}
97C2C_2×\timesC2C_2 (1−18T+pT2)(1−10T+pT2) ( 1 - 18 T + p T^{2} )( 1 - 10 T + p T^{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

−12.40455669710029462914444723137, −11.95540212762440671293789634833, −11.50241564803008008621431292804, −10.96209673656104862541748065987, −10.23323249431693370205575833354, −9.799114190252748412073333481185, −8.962359067748700101158432842733, −8.271522425507048577984329398951, −7.32472320975423953442563815704, −7.00282080312508626852911634054, −6.20992784451237885601763112834, −5.15770247927932104238955318180, −4.79548037719004453433730782907, −3.43778539088294327646502733438, −2.08672222628823078726316203122, 2.08672222628823078726316203122, 3.43778539088294327646502733438, 4.79548037719004453433730782907, 5.15770247927932104238955318180, 6.20992784451237885601763112834, 7.00282080312508626852911634054, 7.32472320975423953442563815704, 8.271522425507048577984329398951, 8.962359067748700101158432842733, 9.799114190252748412073333481185, 10.23323249431693370205575833354, 10.96209673656104862541748065987, 11.50241564803008008621431292804, 11.95540212762440671293789634833, 12.40455669710029462914444723137

Graph of the ZZ-function along the critical line