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Properties

Label 10-591e5-591.590-c0e5-0-1
Degree 1010
Conductor 7.210×10137.210\times 10^{13}
Sign 11
Analytic cond. 0.002232140.00223214
Root an. cond. 0.5430900.543090
Motivic weight 00
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 + 5·3-s − 5-s − 5·6-s − 7-s + 15·9-s + 10-s − 11-s + 14-s − 5·15-s − 17-s − 15·18-s − 19-s − 5·21-s + 22-s + 35·27-s + 5·30-s − 5·33-s + 34-s + 35-s − 37-s + 38-s + 5·42-s − 43-s − 15·45-s − 5·51-s − 35·54-s + ⋯
L(s)  = 1  − 2-s + 5·3-s − 5-s − 5·6-s − 7-s + 15·9-s + 10-s − 11-s + 14-s − 5·15-s − 17-s − 15·18-s − 19-s − 5·21-s + 22-s + 35·27-s + 5·30-s − 5·33-s + 34-s + 35-s − 37-s + 38-s + 5·42-s − 43-s − 15·45-s − 5·51-s − 35·54-s + ⋯

Functional equation

Λ(s)=((35⋅1975)s/2 ΓC(s)5 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 197^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((35⋅1975)s/2 ΓC(s)5 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 197^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1010
Conductor: 35⋅19753^{5} \cdot 197^{5}
Sign: 11
Analytic conductor: 0.002232140.00223214
Root analytic conductor: 0.5430900.543090
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: induced by χ591(590,⋅)\chi_{591} (590, \cdot )
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (10, 35⋅1975, ( :0,0,0,0,0), 1)(10,\ 3^{5} \cdot 197^{5} ,\ ( \ : 0, 0, 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2408645311.240864531
L(12)L(\frac12) ≈\approx 1.2408645311.240864531
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)
bad3C1C_1 (1−T)5 ( 1 - T )^{5}
197C1C_1 (1−T)5 ( 1 - T )^{5}
good2C10C_{10} 1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}
5C10C_{10} 1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}
7C10C_{10} 1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}
11C10C_{10} 1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}
13C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
17C10C_{10} 1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}
19C10C_{10} 1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}
23C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
29C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
31C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
37C10C_{10} 1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}
41C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
43C10C_{10} 1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}
47C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
53C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
59C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
61C10C_{10} 1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}
67C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
71C10C_{10} 1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}
73C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
79C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
83C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
89C10C_{10} 1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}
97C10C_{10} 1+T+T2+T3+T4+T5+T6+T7+T8+T9+T10 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}
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   L(s)=∏p ∏j=110(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−6.90982890423019141803039018365, −6.87179964416648458582562903539, −6.83835463497953047176918864658, −6.32919895151901999340199458956, −6.31538538301541073400568703538, −5.99257724283257570232639670503, −5.42369569191795396236866110225, −5.27126801929357672670714811071, −4.89246384572632318309812830885, −4.88055421624727734066880052540, −4.40036972344417240334454856703, −4.18550701066793006011325118947, −4.16243177774988857354199660172, −4.15553796664477471072882470712, −3.61001271427431092943992928193, −3.46464620374451226359127263808, −3.37579794647086645013551405443, −3.08302653378897140921481583928, −2.87167538293516623059451299215, −2.61041148005977183526182438709, −2.35137057756022469972169375500, −2.31264171519099436316359024194, −1.82708601027781497052058655860, −1.49978973244071203653125346655, −1.33724585627002362314425100504, 1.33724585627002362314425100504, 1.49978973244071203653125346655, 1.82708601027781497052058655860, 2.31264171519099436316359024194, 2.35137057756022469972169375500, 2.61041148005977183526182438709, 2.87167538293516623059451299215, 3.08302653378897140921481583928, 3.37579794647086645013551405443, 3.46464620374451226359127263808, 3.61001271427431092943992928193, 4.15553796664477471072882470712, 4.16243177774988857354199660172, 4.18550701066793006011325118947, 4.40036972344417240334454856703, 4.88055421624727734066880052540, 4.89246384572632318309812830885, 5.27126801929357672670714811071, 5.42369569191795396236866110225, 5.99257724283257570232639670503, 6.31538538301541073400568703538, 6.32919895151901999340199458956, 6.83835463497953047176918864658, 6.87179964416648458582562903539, 6.90982890423019141803039018365

Graph of the ZZ-function along the critical line