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Properties

Label 4-1323e2-1.1-c1e2-0-16
Degree 44
Conductor 17503291750329
Sign 11
Analytic cond. 111.602111.602
Root an. cond. 3.250263.25026
Motivic weight 11
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 + 2·4-s + 5-s + 5·8-s + 10-s + 5·11-s − 5·13-s + 5·16-s + 6·17-s − 2·19-s + 2·20-s + 5·22-s + 3·23-s + 5·25-s − 5·26-s − 29-s + 10·32-s + 6·34-s + 6·37-s − 2·38-s + 5·40-s + 5·41-s + 43-s + 10·44-s + 3·46-s + 5·50-s − 10·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 4-s + 0.447·5-s + 1.76·8-s + 0.316·10-s + 1.50·11-s − 1.38·13-s + 5/4·16-s + 1.45·17-s − 0.458·19-s + 0.447·20-s + 1.06·22-s + 0.625·23-s + 25-s − 0.980·26-s − 0.185·29-s + 1.76·32-s + 1.02·34-s + 0.986·37-s − 0.324·38-s + 0.790·40-s + 0.780·41-s + 0.152·43-s + 1.50·44-s + 0.442·46-s + 0.707·50-s − 1.38·52-s + ⋯

Functional equation

Λ(s)=(1750329s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1750329s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 17503291750329    =    36⋅743^{6} \cdot 7^{4}
Sign: 11
Analytic conductor: 111.602111.602
Root analytic conductor: 3.250263.25026
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1750329, ( :1/2,1/2), 1)(4,\ 1750329,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 6.3160754316.316075431
L(12)L(\frac12) ≈\approx 6.3160754316.316075431
L(32)L(\frac{3}{2}) 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)
bad3 1 1
7 1 1
good2C22C_2^2 1−T−T2−pT3+p2T4 1 - T - T^{2} - p T^{3} + p^{2} T^{4}
5C22C_2^2 1−T−4T2−pT3+p2T4 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4}
11C22C_2^2 1−5T+14T2−5pT3+p2T4 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4}
13C2C_2 (1−2T+pT2)(1+7T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} )
17C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
19C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
23C22C_2^2 1−3T−14T2−3pT3+p2T4 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4}
29C22C_2^2 1+T−28T2+pT3+p2T4 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4}
31C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
37C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
41C22C_2^2 1−5T−16T2−5pT3+p2T4 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4}
43C22C_2^2 1−T−42T2−pT3+p2T4 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4}
47C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
53C2C_2 (1−9T+pT2)2 ( 1 - 9 T + p T^{2} )^{2}
59C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
61C2C_2 (1+T+pT2)(1+13T+pT2) ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} )
67C22C_2^2 1+4T−51T2+4pT3+p2T4 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4}
71C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
73C2C_2 (1+3T+pT2)2 ( 1 + 3 T + p T^{2} )^{2}
79C22C_2^2 1+8T−15T2+8pT3+p2T4 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4}
83C22C_2^2 1−9T−2T2−9pT3+p2T4 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4}
89C2C_2 (1+13T+pT2)2 ( 1 + 13 T + p T^{2} )^{2}
97C22C_2^2 1+9T−16T2+9pT3+p2T4 1 + 9 T - 16 T^{2} + 9 p T^{3} + p^{2} T^{4}
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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

−9.765654098758353579689589013217, −9.586589070570195209519435301502, −9.120325512982210974524711563501, −8.660790446321989972035890209934, −7.917792923176754336091702883245, −7.83251739516913704350720473763, −7.14676001407335559451244264342, −7.02564641711679470513237426715, −6.57975159154352760199619381254, −6.16084346737969298268377596778, −5.49820462400345034607263816226, −5.31943681330000463004881059574, −4.73996282288709150474556684470, −4.12162831739130975987444954383, −4.06224825622680534074913512773, −3.20436812250029263619432161122, −2.62445024600094657434524034121, −2.29332353966590750384064075843, −1.37531438124214562099809327447, −1.09410855423417378425821570018, 1.09410855423417378425821570018, 1.37531438124214562099809327447, 2.29332353966590750384064075843, 2.62445024600094657434524034121, 3.20436812250029263619432161122, 4.06224825622680534074913512773, 4.12162831739130975987444954383, 4.73996282288709150474556684470, 5.31943681330000463004881059574, 5.49820462400345034607263816226, 6.16084346737969298268377596778, 6.57975159154352760199619381254, 7.02564641711679470513237426715, 7.14676001407335559451244264342, 7.83251739516913704350720473763, 7.917792923176754336091702883245, 8.660790446321989972035890209934, 9.120325512982210974524711563501, 9.586589070570195209519435301502, 9.765654098758353579689589013217

Graph of the ZZ-function along the critical line