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Properties

Label 10-1303e5-1303.1302-c0e5-0-0
Degree 1010
Conductor 3.756×10153.756\times 10^{15}
Sign 11
Analytic cond. 0.1162800.116280
Root an. cond. 0.8064000.806400
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 + 5·9-s − 13-s − 5·18-s − 23-s + 5·25-s + 26-s − 41-s + 46-s + 5·49-s − 5·50-s − 61-s − 67-s − 79-s + 15·81-s + 82-s − 83-s − 89-s − 5·98-s − 103-s − 109-s − 113-s − 5·117-s + 5·121-s + 122-s + 127-s + 131-s + ⋯
L(s)  = 1  − 2-s + 5·9-s − 13-s − 5·18-s − 23-s + 5·25-s + 26-s − 41-s + 46-s + 5·49-s − 5·50-s − 61-s − 67-s − 79-s + 15·81-s + 82-s − 83-s − 89-s − 5·98-s − 103-s − 109-s − 113-s − 5·117-s + 5·121-s + 122-s + 127-s + 131-s + ⋯

Functional equation

Λ(s)=((13035)s/2 ΓC(s)5 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(1303^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((13035)s/2 ΓC(s)5 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(1303^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1010
Conductor: 130351303^{5}
Sign: 11
Analytic conductor: 0.1162800.116280
Root analytic conductor: 0.8064000.806400
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: induced by χ1303(1302,⋅)\chi_{1303} (1302, \cdot )
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (10, 13035, ( :0,0,0,0,0), 1)(10,\ 1303^{5} ,\ ( \ : 0, 0, 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.97501234530.9750123453
L(12)L(\frac12) ≈\approx 0.97501234530.9750123453
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)
bad1303C1C_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}
3C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
5C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
7C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
11C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
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}
19C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
23C10C_{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}
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}
37C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
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}
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}
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}
71C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
73C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
79C10C_{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}
83C10C_{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}
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}
97C1C_1×\timesC1C_1 (1−T)5(1+T)5 ( 1 - T )^{5}( 1 + T )^{5}
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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

−6.15906246070400453764973182937, −5.95610174947768459102177880246, −5.59781757676187892478949981359, −5.38764584205613917884399296923, −5.38701991796606133332280685529, −4.86799811902599925316447841510, −4.77573403747939430720728191678, −4.68728365651808918880788767128, −4.61905077612086876373932789728, −4.60827826990408734939009745928, −4.04644317575707131527640531549, −3.95552809636614571698924215850, −3.91722517471613998744552799378, −3.81750756496842638922539402069, −3.33890072597395085145558201184, −3.06814001176919337519022237188, −2.89525105345502872843806324297, −2.65740367150367347755910876893, −2.20984866761775695625174137821, −2.18553531301260884791012770538, −1.97461115041838847854311288998, −1.41163479920336533613124313004, −1.11150612389930148990524845605, −1.09945564540253261297300848596, −1.08036710951809838189526682606, 1.08036710951809838189526682606, 1.09945564540253261297300848596, 1.11150612389930148990524845605, 1.41163479920336533613124313004, 1.97461115041838847854311288998, 2.18553531301260884791012770538, 2.20984866761775695625174137821, 2.65740367150367347755910876893, 2.89525105345502872843806324297, 3.06814001176919337519022237188, 3.33890072597395085145558201184, 3.81750756496842638922539402069, 3.91722517471613998744552799378, 3.95552809636614571698924215850, 4.04644317575707131527640531549, 4.60827826990408734939009745928, 4.61905077612086876373932789728, 4.68728365651808918880788767128, 4.77573403747939430720728191678, 4.86799811902599925316447841510, 5.38701991796606133332280685529, 5.38764584205613917884399296923, 5.59781757676187892478949981359, 5.95610174947768459102177880246, 6.15906246070400453764973182937

Graph of the ZZ-function along the critical line