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Properties

Label 4-624e2-1.1-c3e2-0-3
Degree 44
Conductor 389376389376
Sign 11
Analytic cond. 1355.501355.50
Root an. cond. 6.067716.06771
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s + 27·9-s − 26·13-s + 36·17-s − 48·23-s + 202·25-s − 108·27-s + 12·29-s + 156·39-s − 40·43-s + 578·49-s − 216·51-s − 612·53-s + 140·61-s + 288·69-s − 1.21e3·75-s + 832·79-s + 405·81-s − 72·87-s + 2.55e3·101-s + 1.75e3·107-s + 1.38e3·113-s − 702·117-s + 2.36e3·121-s + 127-s + 240·129-s + 131-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s − 0.554·13-s + 0.513·17-s − 0.435·23-s + 1.61·25-s − 0.769·27-s + 0.0768·29-s + 0.640·39-s − 0.141·43-s + 1.68·49-s − 0.593·51-s − 1.58·53-s + 0.293·61-s + 0.502·69-s − 1.86·75-s + 1.18·79-s + 5/9·81-s − 0.0887·87-s + 2.51·101-s + 1.58·107-s + 1.14·113-s − 0.554·117-s + 1.77·121-s + 0.000698·127-s + 0.163·129-s + 0.000666·131-s + ⋯

Functional equation

Λ(s)=(389376s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(389376s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 389376389376    =    28⋅32⋅1322^{8} \cdot 3^{2} \cdot 13^{2}
Sign: 11
Analytic conductor: 1355.501355.50
Root analytic conductor: 6.067716.06771
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 389376, ( :3/2,3/2), 1)(4,\ 389376,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.5662201461.566220146
L(12)L(\frac12) ≈\approx 1.5662201461.566220146
L(52)L(\frac{5}{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)
bad2 1 1
3C1C_1 (1+pT)2 ( 1 + p T )^{2}
13C2C_2 1+2pT+p3T2 1 + 2 p T + p^{3} T^{2}
good5C22C_2^2 1−202T2+p6T4 1 - 202 T^{2} + p^{6} T^{4}
7C22C_2^2 1−578T2+p6T4 1 - 578 T^{2} + p^{6} T^{4}
11C22C_2^2 1−2362T2+p6T4 1 - 2362 T^{2} + p^{6} T^{4}
17C2C_2 (1−18T+p3T2)2 ( 1 - 18 T + p^{3} T^{2} )^{2}
19C22C_2^2 1−11690T2+p6T4 1 - 11690 T^{2} + p^{6} T^{4}
23C2C_2 (1+24T+p3T2)2 ( 1 + 24 T + p^{3} T^{2} )^{2}
29C2C_2 (1−6T+p3T2)2 ( 1 - 6 T + p^{3} T^{2} )^{2}
31C22C_2^2 1−50p2T2+p6T4 1 - 50 p^{2} T^{2} + p^{6} T^{4}
37C22C_2^2 1−66314T2+p6T4 1 - 66314 T^{2} + p^{6} T^{4}
41C22C_2^2 1−125554T2+p6T4 1 - 125554 T^{2} + p^{6} T^{4}
43C2C_2 (1+20T+p3T2)2 ( 1 + 20 T + p^{3} T^{2} )^{2}
47C22C_2^2 1−120946T2+p6T4 1 - 120946 T^{2} + p^{6} T^{4}
53C2C_2 (1+306T+p3T2)2 ( 1 + 306 T + p^{3} T^{2} )^{2}
59C22C_2^2 1+83750T2+p6T4 1 + 83750 T^{2} + p^{6} T^{4}
61C2C_2 (1−70T+p3T2)2 ( 1 - 70 T + p^{3} T^{2} )^{2}
67C22C_2^2 1−395594T2+p6T4 1 - 395594 T^{2} + p^{6} T^{4}
71C22C_2^2 1+342686T2+p6T4 1 + 342686 T^{2} + p^{6} T^{4}
73C22C_2^2 1−777842T2+p6T4 1 - 777842 T^{2} + p^{6} T^{4}
79C2C_2 (1−416T+p3T2)2 ( 1 - 416 T + p^{3} T^{2} )^{2}
83C22C_2^2 1+17078T2+p6T4 1 + 17078 T^{2} + p^{6} T^{4}
89C22C_2^2 1−1409170T2+p6T4 1 - 1409170 T^{2} + p^{6} T^{4}
97C22C_2^2 1−1548098T2+p6T4 1 - 1548098 T^{2} + p^{6} 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

−10.49361312444175652880191203906, −10.08233209661012421080280868651, −9.691089477507971603465867743613, −9.204462708357827299556523815611, −8.686421092017824869469162151228, −8.277717279890582151086819338876, −7.48944476379671863725420031909, −7.41149842440090257512723258767, −6.81854566334900337736092477414, −6.26354100113121662877903392870, −5.99428538010790908656445017488, −5.37400025855853906992394171624, −4.81767351197160746903693914730, −4.69823802741809431962518271939, −3.85566403046489682942511542100, −3.31735195520040350489574850926, −2.57538884598504334606969270529, −1.85941798453617844214838396428, −1.02127159714424799426160220637, −0.47359552811597610726077333166, 0.47359552811597610726077333166, 1.02127159714424799426160220637, 1.85941798453617844214838396428, 2.57538884598504334606969270529, 3.31735195520040350489574850926, 3.85566403046489682942511542100, 4.69823802741809431962518271939, 4.81767351197160746903693914730, 5.37400025855853906992394171624, 5.99428538010790908656445017488, 6.26354100113121662877903392870, 6.81854566334900337736092477414, 7.41149842440090257512723258767, 7.48944476379671863725420031909, 8.277717279890582151086819338876, 8.686421092017824869469162151228, 9.204462708357827299556523815611, 9.691089477507971603465867743613, 10.08233209661012421080280868651, 10.49361312444175652880191203906

Graph of the ZZ-function along the critical line