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Properties

Label 4-425e2-1.1-c3e2-0-7
Degree 44
Conductor 180625180625
Sign 11
Analytic cond. 628.796628.796
Root an. cond. 5.007575.00757
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 7·4-s − 10·9-s − 48·11-s − 15·16-s − 232·19-s − 60·29-s − 344·31-s − 70·36-s − 684·41-s − 336·44-s − 98·49-s − 504·59-s + 220·61-s − 553·64-s − 1.41e3·71-s − 1.62e3·76-s + 968·79-s − 629·81-s + 1.54e3·89-s + 480·99-s − 420·101-s + 2.37e3·109-s − 420·116-s − 934·121-s − 2.40e3·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 7/8·4-s − 0.370·9-s − 1.31·11-s − 0.234·16-s − 2.80·19-s − 0.384·29-s − 1.99·31-s − 0.324·36-s − 2.60·41-s − 1.15·44-s − 2/7·49-s − 1.11·59-s + 0.461·61-s − 1.08·64-s − 2.36·71-s − 2.45·76-s + 1.37·79-s − 0.862·81-s + 1.84·89-s + 0.487·99-s − 0.413·101-s + 2.08·109-s − 0.336·116-s − 0.701·121-s − 1.74·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

Functional equation

Λ(s)=(180625s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(180625s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 180625180625    =    54⋅1725^{4} \cdot 17^{2}
Sign: 11
Analytic conductor: 628.796628.796
Root analytic conductor: 5.007575.00757
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 180625, ( :3/2,3/2), 1)(4,\ 180625,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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)
bad5 1 1
17C2C_2 1+p2T2 1 + p^{2} T^{2}
good2C22C_2^2 1−7T2+p6T4 1 - 7 T^{2} + p^{6} T^{4}
3C22C_2^2 1+10T2+p6T4 1 + 10 T^{2} + p^{6} T^{4}
7C22C_2^2 1+2p2T2+p6T4 1 + 2 p^{2} T^{2} + p^{6} T^{4}
11C2C_2 (1+24T+p3T2)2 ( 1 + 24 T + p^{3} T^{2} )^{2}
13C22C_2^2 1−1030T2+p6T4 1 - 1030 T^{2} + p^{6} T^{4}
19C2C_2 (1+116T+p3T2)2 ( 1 + 116 T + p^{3} T^{2} )^{2}
23C22C_2^2 1−20734T2+p6T4 1 - 20734 T^{2} + p^{6} T^{4}
29C2C_2 (1+30T+p3T2)2 ( 1 + 30 T + p^{3} T^{2} )^{2}
31C2C_2 (1+172T+p3T2)2 ( 1 + 172 T + p^{3} T^{2} )^{2}
37C22C_2^2 1−97942T2+p6T4 1 - 97942 T^{2} + p^{6} T^{4}
41C2C_2 (1+342T+p3T2)2 ( 1 + 342 T + p^{3} T^{2} )^{2}
43C22C_2^2 1−137110T2+p6T4 1 - 137110 T^{2} + p^{6} T^{4}
47C22C_2^2 1−124702T2+p6T4 1 - 124702 T^{2} + p^{6} T^{4}
53C22C_2^2 1−70p2T2+p6T4 1 - 70 p^{2} T^{2} + p^{6} T^{4}
59C2C_2 (1+252T+p3T2)2 ( 1 + 252 T + p^{3} T^{2} )^{2}
61C2C_2 (1−110T+p3T2)2 ( 1 - 110 T + p^{3} T^{2} )^{2}
67C22C_2^2 1−367270T2+p6T4 1 - 367270 T^{2} + p^{6} T^{4}
71C2C_2 (1+708T+p3T2)2 ( 1 + 708 T + p^{3} T^{2} )^{2}
73C22C_2^2 1−646990T2+p6T4 1 - 646990 T^{2} + p^{6} T^{4}
79C2C_2 (1−484T+p3T2)2 ( 1 - 484 T + p^{3} T^{2} )^{2}
83C22C_2^2 1−572038T2+p6T4 1 - 572038 T^{2} + p^{6} T^{4}
89C2C_2 (1−774T+p3T2)2 ( 1 - 774 T + p^{3} T^{2} )^{2}
97C22C_2^2 1−1679422T2+p6T4 1 - 1679422 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

−10.48262812718079295191751081100, −10.45003594599013903395813572461, −9.748953406935180400184091027290, −8.916466472894863703065734888805, −8.772156140594064990213604724660, −8.317040883954338335083503122259, −7.52430748825825200745536874118, −7.48714667875867588271705789486, −6.61879848170905525825960160666, −6.41697704808275461280133467235, −5.84798190277569519804114226122, −5.21524494767023274350368859995, −4.76542088483137001492988901804, −4.06550794064444127682719866466, −3.37760450585846030072698215390, −2.73931685831830878603720113640, −2.00323208346037421144177330888, −1.81424147318915866618309580227, 0, 0, 1.81424147318915866618309580227, 2.00323208346037421144177330888, 2.73931685831830878603720113640, 3.37760450585846030072698215390, 4.06550794064444127682719866466, 4.76542088483137001492988901804, 5.21524494767023274350368859995, 5.84798190277569519804114226122, 6.41697704808275461280133467235, 6.61879848170905525825960160666, 7.48714667875867588271705789486, 7.52430748825825200745536874118, 8.317040883954338335083503122259, 8.772156140594064990213604724660, 8.916466472894863703065734888805, 9.748953406935180400184091027290, 10.45003594599013903395813572461, 10.48262812718079295191751081100

Graph of the ZZ-function along the critical line