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Properties

Label 4-1519e2-1.1-c0e2-0-0
Degree 44
Conductor 23073612307361
Sign 11
Analytic cond. 0.5746840.574684
Root an. cond. 0.8706770.870677
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 + 4-s − 5-s + 2·8-s − 9-s − 10-s + 2·16-s − 18-s − 19-s − 20-s + 25-s + 31-s + 2·32-s − 36-s − 38-s − 2·40-s + 2·41-s + 45-s + 2·47-s + 50-s − 59-s + 62-s + 3·64-s − 2·67-s − 2·71-s − 2·72-s − 76-s + ⋯
L(s)  = 1  + 2-s + 4-s − 5-s + 2·8-s − 9-s − 10-s + 2·16-s − 18-s − 19-s − 20-s + 25-s + 31-s + 2·32-s − 36-s − 38-s − 2·40-s + 2·41-s + 45-s + 2·47-s + 50-s − 59-s + 62-s + 3·64-s − 2·67-s − 2·71-s − 2·72-s − 76-s + ⋯

Functional equation

Λ(s)=(2307361s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2307361 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(2307361s/2 ΓC(s)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2307361 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 23073612307361    =    74⋅3127^{4} \cdot 31^{2}
Sign: 11
Analytic conductor: 0.5746840.574684
Root analytic conductor: 0.8706770.870677
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 2307361, ( :0,0), 1)(4,\ 2307361,\ (\ :0, 0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.9024287381.902428738
L(12)L(\frac12) ≈\approx 1.9024287381.902428738
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)
bad7 1 1
31C2C_2 1−T+T2 1 - T + T^{2}
good2C1C_1×\timesC2C_2 (1−T)2(1+T+T2) ( 1 - T )^{2}( 1 + T + T^{2} )
3C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
5C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
11C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
13C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
17C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
19C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
23C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
29C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
37C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
41C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
43C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
47C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{2}
53C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
59C1C_1×\timesC2C_2 (1+T)2(1−T+T2) ( 1 + T )^{2}( 1 - T + T^{2} )
61C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
67C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
71C2C_2 (1+T+T2)2 ( 1 + T + T^{2} )^{2}
73C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
79C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
83C1C_1×\timesC1C_1 (1−T)2(1+T)2 ( 1 - T )^{2}( 1 + T )^{2}
89C2C_2 (1−T+T2)(1+T+T2) ( 1 - T + T^{2} )( 1 + T + T^{2} )
97C2C_2 (1−T+T2)2 ( 1 - T + T^{2} )^{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

−10.18120124247099726860934638272, −9.356040631438886640778695076989, −8.860705203322919815819196350857, −8.743879257593704398437238735820, −7.944948598441943294811610760986, −7.906186467820005687428357321296, −7.42579268416548263474977017681, −7.14542225264980719802034565323, −6.48425072733216374184969933383, −6.22421965381850730879239178216, −5.68019894342069656689627642347, −5.37159574964039234795200812692, −4.57982136357682714328017470306, −4.41773479427198656877679632591, −4.14707869902630960258815201170, −3.51969767269653018714950428693, −2.78809519777782555222884992936, −2.68790291850424569650232601203, −1.84312791933419256143066304880, −1.00892909238363937758976610376, 1.00892909238363937758976610376, 1.84312791933419256143066304880, 2.68790291850424569650232601203, 2.78809519777782555222884992936, 3.51969767269653018714950428693, 4.14707869902630960258815201170, 4.41773479427198656877679632591, 4.57982136357682714328017470306, 5.37159574964039234795200812692, 5.68019894342069656689627642347, 6.22421965381850730879239178216, 6.48425072733216374184969933383, 7.14542225264980719802034565323, 7.42579268416548263474977017681, 7.906186467820005687428357321296, 7.944948598441943294811610760986, 8.743879257593704398437238735820, 8.860705203322919815819196350857, 9.356040631438886640778695076989, 10.18120124247099726860934638272

Graph of the ZZ-function along the critical line