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Properties

Label 10-1304e5-1304.325-c0e5-0-1
Degree 1010
Conductor 3.770×10153.770\times 10^{15}
Sign 11
Analytic cond. 0.1167270.116727
Root an. cond. 0.8067090.806709
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  + 5·2-s − 3-s + 15·4-s − 5-s − 5·6-s + 35·8-s − 5·10-s − 11-s − 15·12-s − 13-s + 15-s + 70·16-s − 19-s − 15·20-s − 5·22-s − 35·24-s − 5·26-s − 29-s + 5·30-s + 126·32-s + 33-s − 37-s − 5·38-s + 39-s − 35·40-s − 41-s − 15·44-s + ⋯
L(s)  = 1  + 5·2-s − 3-s + 15·4-s − 5-s − 5·6-s + 35·8-s − 5·10-s − 11-s − 15·12-s − 13-s + 15-s + 70·16-s − 19-s − 15·20-s − 5·22-s − 35·24-s − 5·26-s − 29-s + 5·30-s + 126·32-s + 33-s − 37-s − 5·38-s + 39-s − 35·40-s − 41-s − 15·44-s + ⋯

Functional equation

Λ(s)=((215⋅1635)s/2 ΓC(s)5 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 163^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((215⋅1635)s/2 ΓC(s)5 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 163^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1010
Conductor: 215⋅16352^{15} \cdot 163^{5}
Sign: 11
Analytic conductor: 0.1167270.116727
Root analytic conductor: 0.8067090.806709
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: induced by χ1304(325,⋅)\chi_{1304} (325, \cdot )
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (10, 215⋅1635, ( :0,0,0,0,0), 1)(10,\ 2^{15} \cdot 163^{5} ,\ ( \ : 0, 0, 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 17.9935438417.99354384
L(12)L(\frac12) ≈\approx 17.9935438417.99354384
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)5 ( 1 - T )^{5}
163C1C_1 (1−T)5 ( 1 - T )^{5}
good3C10C_{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}
7C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
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}
13C10C_{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}
17C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
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}
29C10C_{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}
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}
41C10C_{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}
43C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
47C10C_{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}
53C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
59C10C_{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}
61C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
67C10C_{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}
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}
89C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
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

−5.69550458013521488575610389028, −5.67339253879251392499874973422, −5.58711839830927816182622788364, −5.45369651440744799427916734818, −5.26986216128255403070173974495, −5.16205928780752946834742037975, −5.12511175921664206642588528737, −4.59163045550744044033544825358, −4.54744526291529868957369200104, −4.38678492240882684908836490886, −4.24103529500060714738287590218, −4.04191907951308337052558724396, −3.94200282677566409564013445744, −3.78860019094056842198598593270, −3.43149195583217359391445643443, −3.22506122010674913647964370861, −3.12116726266440893801343968522, −2.76287056834018055160439405415, −2.69534322685201740588308474433, −2.55433929013862000853122646330, −2.05378594073999999068355458779, −1.97478062792938487520406885512, −1.93455060978440383216460627740, −1.34508569576849823377576793347, −1.11302435206381325298402161645, 1.11302435206381325298402161645, 1.34508569576849823377576793347, 1.93455060978440383216460627740, 1.97478062792938487520406885512, 2.05378594073999999068355458779, 2.55433929013862000853122646330, 2.69534322685201740588308474433, 2.76287056834018055160439405415, 3.12116726266440893801343968522, 3.22506122010674913647964370861, 3.43149195583217359391445643443, 3.78860019094056842198598593270, 3.94200282677566409564013445744, 4.04191907951308337052558724396, 4.24103529500060714738287590218, 4.38678492240882684908836490886, 4.54744526291529868957369200104, 4.59163045550744044033544825358, 5.12511175921664206642588528737, 5.16205928780752946834742037975, 5.26986216128255403070173974495, 5.45369651440744799427916734818, 5.58711839830927816182622788364, 5.67339253879251392499874973422, 5.69550458013521488575610389028

Graph of the ZZ-function along the critical line