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Properties

Label 6-3800e3-152.37-c0e3-0-0
Degree 66
Conductor 5487200000054872000000
Sign 11
Analytic cond. 6.820596.82059
Root an. cond. 1.377111.37711
Motivic weight 00
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  − 3·2-s + 6·4-s − 10·8-s + 15·16-s − 3·19-s + 27-s − 21·32-s + 3·37-s + 9·38-s − 3·47-s − 3·54-s + 28·64-s − 9·74-s − 18·76-s + 9·94-s + 6·108-s + 3·121-s + 127-s − 36·128-s + 131-s + 137-s + 139-s + 18·148-s + 149-s + 151-s + 30·152-s + 157-s + ⋯
L(s)  = 1  − 3·2-s + 6·4-s − 10·8-s + 15·16-s − 3·19-s + 27-s − 21·32-s + 3·37-s + 9·38-s − 3·47-s − 3·54-s + 28·64-s − 9·74-s − 18·76-s + 9·94-s + 6·108-s + 3·121-s + 127-s − 36·128-s + 131-s + 137-s + 139-s + 18·148-s + 149-s + 151-s + 30·152-s + 157-s + ⋯

Functional equation

Λ(s)=((29⋅56⋅193)s/2 ΓC(s)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((29⋅56⋅193)s/2 ΓC(s)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 29⋅56⋅1932^{9} \cdot 5^{6} \cdot 19^{3}
Sign: 11
Analytic conductor: 6.820596.82059
Root analytic conductor: 1.377111.37711
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: induced by χ3800(1101,⋅)\chi_{3800} (1101, \cdot )
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 29⋅56⋅193, ( :0,0,0), 1)(6,\ 2^{9} \cdot 5^{6} \cdot 19^{3} ,\ ( \ : 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.31823385730.3182338573
L(12)L(\frac12) ≈\approx 0.31823385730.3182338573
L(1)L(1) 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)
bad2C1C_1 (1+T)3 ( 1 + T )^{3}
5 1 1
19C1C_1 (1+T)3 ( 1 + T )^{3}
good3C6C_6 1−T3+T6 1 - T^{3} + T^{6}
7C6C_6 1+T3+T6 1 + T^{3} + T^{6}
11C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
13C6C_6 1−T3+T6 1 - T^{3} + T^{6}
17C6C_6 1+T3+T6 1 + T^{3} + T^{6}
23C6C_6 1+T3+T6 1 + T^{3} + T^{6}
29C6C_6 1−T3+T6 1 - T^{3} + T^{6}
31C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
37C2C_2 (1−T+T2)3 ( 1 - T + T^{2} )^{3}
41C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
43C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
47C2C_2 (1+T+T2)3 ( 1 + T + T^{2} )^{3}
53C6C_6 1−T3+T6 1 - T^{3} + T^{6}
59C6C_6 1−T3+T6 1 - T^{3} + T^{6}
61C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
67C6C_6 1−T3+T6 1 - T^{3} + T^{6}
71C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
73C6C_6 1+T3+T6 1 + T^{3} + T^{6}
79C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
83C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
89C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
97C1C_1×\timesC1C_1 (1−T)3(1+T)3 ( 1 - T )^{3}( 1 + T )^{3}
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   L(s)=∏p ∏j=16(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.920433166717209397823752313459, −7.74696446579968495356410257820, −7.22259940738282807331510472649, −6.94738330976620738177121407657, −6.93300801448237260386046237234, −6.57910072727121896937510160842, −6.32276338722519711534820452520, −6.23220699731047492022741219470, −5.89163009425620121811726394799, −5.87858761736417789078448667577, −5.29325736723530133678251333539, −4.98360230009489519366366666265, −4.61408884470067770113623007625, −4.33720108578422471704685902859, −3.91515814680727749871202078643, −3.74698871582306586024589522387, −3.01541659037606063468058618145, −2.97525884109578343826211530157, −2.84304511510068920946380533510, −2.29358438037025803274176720683, −2.03319230706837636392119405231, −1.73590910442619845156342712456, −1.55059409854462001210166804268, −0.790026702321987899437065594714, −0.53475195151724166465600046100, 0.53475195151724166465600046100, 0.790026702321987899437065594714, 1.55059409854462001210166804268, 1.73590910442619845156342712456, 2.03319230706837636392119405231, 2.29358438037025803274176720683, 2.84304511510068920946380533510, 2.97525884109578343826211530157, 3.01541659037606063468058618145, 3.74698871582306586024589522387, 3.91515814680727749871202078643, 4.33720108578422471704685902859, 4.61408884470067770113623007625, 4.98360230009489519366366666265, 5.29325736723530133678251333539, 5.87858761736417789078448667577, 5.89163009425620121811726394799, 6.23220699731047492022741219470, 6.32276338722519711534820452520, 6.57910072727121896937510160842, 6.93300801448237260386046237234, 6.94738330976620738177121407657, 7.22259940738282807331510472649, 7.74696446579968495356410257820, 7.920433166717209397823752313459

Graph of the ZZ-function along the critical line