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Properties

Label 4-175e2-1.1-c9e2-0-2
Degree 44
Conductor 3062530625
Sign 11
Analytic cond. 8123.648123.64
Root an. cond. 9.493749.49374
Motivic weight 99
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 24·2-s + 174·3-s − 584·4-s + 4.17e3·6-s − 4.80e3·7-s − 2.95e4·8-s + 6.66e3·9-s + 1.85e4·11-s − 1.01e5·12-s + 5.10e4·13-s − 1.15e5·14-s + 576·16-s + 3.73e5·17-s + 1.60e5·18-s − 1.43e5·19-s − 8.35e5·21-s + 4.45e5·22-s + 4.98e5·23-s − 5.14e6·24-s + 1.22e6·26-s + 4.77e5·27-s + 2.80e6·28-s − 1.15e7·29-s − 3.95e6·31-s + 2.08e7·32-s + 3.23e6·33-s + 8.97e6·34-s + ⋯
L(s)  = 1  + 1.06·2-s + 1.24·3-s − 1.14·4-s + 1.31·6-s − 0.755·7-s − 2.55·8-s + 0.338·9-s + 0.382·11-s − 1.41·12-s + 0.496·13-s − 0.801·14-s + 0.00219·16-s + 1.08·17-s + 0.359·18-s − 0.252·19-s − 0.937·21-s + 0.405·22-s + 0.371·23-s − 3.16·24-s + 0.526·26-s + 0.172·27-s + 0.862·28-s − 3.03·29-s − 0.768·31-s + 3.51·32-s + 0.474·33-s + 1.15·34-s + ⋯

Functional equation

Λ(s)=(30625s/2 ΓC(s)2 L(s)=( Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
Λ(s)=(30625s/2 ΓC(s+9/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 3062530625    =    54⋅725^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 8123.648123.64
Root analytic conductor: 9.493749.49374
Motivic weight: 99
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 30625, ( :9/2,9/2), 1)(4,\ 30625,\ (\ :9/2, 9/2),\ 1)

Particular Values

L(5)L(5) == 00
L(12)L(\frac12) == 00
L(112)L(\frac{11}{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
7C1C_1 (1+p4T)2 ( 1 + p^{4} T )^{2}
good2D4D_{4} 1−3p3T+145p3T2−3p12T3+p18T4 1 - 3 p^{3} T + 145 p^{3} T^{2} - 3 p^{12} T^{3} + p^{18} T^{4}
3D4D_{4} 1−58pT+2623p2T2−58p10T3+p18T4 1 - 58 p T + 2623 p^{2} T^{2} - 58 p^{10} T^{3} + p^{18} T^{4}
11D4D_{4} 1−18566T+4136090463T2−18566p9T3+p18T4 1 - 18566 T + 4136090463 T^{2} - 18566 p^{9} T^{3} + p^{18} T^{4}
13D4D_{4} 1−3930pT+19172461323T2−3930p10T3+p18T4 1 - 3930 p T + 19172461323 T^{2} - 3930 p^{10} T^{3} + p^{18} T^{4}
17D4D_{4} 1−373910T+270283008251T2−373910p9T3+p18T4 1 - 373910 T + 270283008251 T^{2} - 373910 p^{9} T^{3} + p^{18} T^{4}
19D4D_{4} 1+143276T+642750798250T2+143276p9T3+p18T4 1 + 143276 T + 642750798250 T^{2} + 143276 p^{9} T^{3} + p^{18} T^{4}
23D4D_{4} 1−498908T−1733603053870T2−498908p9T3+p18T4 1 - 498908 T - 1733603053870 T^{2} - 498908 p^{9} T^{3} + p^{18} T^{4}
29D4D_{4} 1+399226pT+61819642147595T2+399226p10T3+p18T4 1 + 399226 p T + 61819642147595 T^{2} + 399226 p^{10} T^{3} + p^{18} T^{4}
31D4D_{4} 1+3953760T+18380996896542T2+3953760p9T3+p18T4 1 + 3953760 T + 18380996896542 T^{2} + 3953760 p^{9} T^{3} + p^{18} T^{4}
37D4D_{4} 1−3205412T+134679597433390T2−3205412p9T3+p18T4 1 - 3205412 T + 134679597433390 T^{2} - 3205412 p^{9} T^{3} + p^{18} T^{4}
41D4D_{4} 1−1058992T−202350146478094T2−1058992p9T3+p18T4 1 - 1058992 T - 202350146478094 T^{2} - 1058992 p^{9} T^{3} + p^{18} T^{4}
43D4D_{4} 1+15948180T+312060834724314T2+15948180p9T3+p18T4 1 + 15948180 T + 312060834724314 T^{2} + 15948180 p^{9} T^{3} + p^{18} T^{4}
47D4D_{4} 1+65501290T+2591944660543287T2+65501290p9T3+p18T4 1 + 65501290 T + 2591944660543287 T^{2} + 65501290 p^{9} T^{3} + p^{18} T^{4}
53D4D_{4} 1−25114688T+6536128371514410T2−25114688p9T3+p18T4 1 - 25114688 T + 6536128371514410 T^{2} - 25114688 p^{9} T^{3} + p^{18} T^{4}
59D4D_{4} 1+116159208T+8473255943386694T2+116159208p9T3+p18T4 1 + 116159208 T + 8473255943386694 T^{2} + 116159208 p^{9} T^{3} + p^{18} T^{4}
61D4D_{4} 1+44688544T+21891928402164378T2+44688544p9T3+p18T4 1 + 44688544 T + 21891928402164378 T^{2} + 44688544 p^{9} T^{3} + p^{18} T^{4}
67D4D_{4} 1+118092496T+11543378662237830T2+118092496p9T3+p18T4 1 + 118092496 T + 11543378662237830 T^{2} + 118092496 p^{9} T^{3} + p^{18} T^{4}
71D4D_{4} 1+294165824T+66419314231297806T2+294165824p9T3+p18T4 1 + 294165824 T + 66419314231297806 T^{2} + 294165824 p^{9} T^{3} + p^{18} T^{4}
73D4D_{4} 1−57419332T+116103868936695574T2−57419332p9T3+p18T4 1 - 57419332 T + 116103868936695574 T^{2} - 57419332 p^{9} T^{3} + p^{18} T^{4}
79D4D_{4} 1+692852854T+353229306838520119T2+692852854p9T3+p18T4 1 + 692852854 T + 353229306838520119 T^{2} + 692852854 p^{9} T^{3} + p^{18} T^{4}
83D4D_{4} 1−6514216pT+378894185867322102T2−6514216p10T3+p18T4 1 - 6514216 p T + 378894185867322102 T^{2} - 6514216 p^{10} T^{3} + p^{18} T^{4}
89D4D_{4} 1+779043704T+776620283810146850T2+779043704p9T3+p18T4 1 + 779043704 T + 776620283810146850 T^{2} + 779043704 p^{9} T^{3} + p^{18} T^{4}
97D4D_{4} 1−2673039406T+3290052660205451443T2−2673039406p9T3+p18T4 1 - 2673039406 T + 3290052660205451443 T^{2} - 2673039406 p^{9} T^{3} + p^{18} 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

−10.72332263277543635520212718063, −10.04503603989193165192675669081, −9.516681435355276758194566916491, −9.248424858875174770972139314890, −8.854443681950545430007021630017, −8.439618704201658616730309079442, −7.78180129729300399453928843940, −7.31599339660262361090232450626, −6.29753536532339788144506532357, −5.95557579400063811242387947680, −5.36274064545844210088903213003, −4.84506123048857842154839867913, −3.98796071192884477735336646048, −3.75261804553506790169688313550, −3.09407160313781715151718558873, −3.05288156102244809378056457542, −1.86255070794891609193750177491, −1.13743857261380907854333089593, 0, 0, 1.13743857261380907854333089593, 1.86255070794891609193750177491, 3.05288156102244809378056457542, 3.09407160313781715151718558873, 3.75261804553506790169688313550, 3.98796071192884477735336646048, 4.84506123048857842154839867913, 5.36274064545844210088903213003, 5.95557579400063811242387947680, 6.29753536532339788144506532357, 7.31599339660262361090232450626, 7.78180129729300399453928843940, 8.439618704201658616730309079442, 8.854443681950545430007021630017, 9.248424858875174770972139314890, 9.516681435355276758194566916491, 10.04503603989193165192675669081, 10.72332263277543635520212718063

Graph of the ZZ-function along the critical line