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Properties

Label 4-2175e2-1.1-c1e2-0-5
Degree 44
Conductor 47306254730625
Sign 11
Analytic cond. 301.628301.628
Root an. cond. 4.167424.16742
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 − 2·3-s + 4-s + 2·6-s − 2·7-s − 3·8-s + 3·9-s − 7·11-s − 2·12-s + 4·13-s + 2·14-s + 16-s + 6·17-s − 3·18-s + 2·19-s + 4·21-s + 7·22-s − 9·23-s + 6·24-s − 4·26-s − 4·27-s − 2·28-s + 2·29-s + 8·31-s + 32-s + 14·33-s − 6·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.755·7-s − 1.06·8-s + 9-s − 2.11·11-s − 0.577·12-s + 1.10·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.707·18-s + 0.458·19-s + 0.872·21-s + 1.49·22-s − 1.87·23-s + 1.22·24-s − 0.784·26-s − 0.769·27-s − 0.377·28-s + 0.371·29-s + 1.43·31-s + 0.176·32-s + 2.43·33-s − 1.02·34-s + ⋯

Functional equation

Λ(s)=(4730625s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4730625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4730625s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4730625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 47306254730625    =    32⋅54⋅2923^{2} \cdot 5^{4} \cdot 29^{2}
Sign: 11
Analytic conductor: 301.628301.628
Root analytic conductor: 4.167424.16742
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 4730625, ( :1/2,1/2), 1)(4,\ 4730625,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.84843260100.8484326010
L(12)L(\frac12) ≈\approx 0.84843260100.8484326010
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)
bad3C1C_1 (1+T)2 ( 1 + T )^{2}
5 1 1
29C1C_1 (1−T)2 ( 1 - T )^{2}
good2D4D_{4} 1+T+pT3+p2T4 1 + T + p T^{3} + p^{2} T^{4}
7D4D_{4} 1+2T−2T2+2pT3+p2T4 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4}
11D4D_{4} 1+7T+30T2+7pT3+p2T4 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4}
13C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
17D4D_{4} 1−6T+26T2−6pT3+p2T4 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4}
19D4D_{4} 1−2T+22T2−2pT3+p2T4 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+9T+62T2+9pT3+p2T4 1 + 9 T + 62 T^{2} + 9 p T^{3} + p^{2} T^{4}
31C2C_2 (1−4T+pT2)2 ( 1 - 4 T + p T^{2} )^{2}
37D4D_{4} 1+9T+56T2+9pT3+p2T4 1 + 9 T + 56 T^{2} + 9 p T^{3} + p^{2} T^{4}
41D4D_{4} 1−9T+64T2−9pT3+p2T4 1 - 9 T + 64 T^{2} - 9 p T^{3} + p^{2} T^{4}
43D4D_{4} 1+3T+50T2+3pT3+p2T4 1 + 3 T + 50 T^{2} + 3 p T^{3} + p^{2} T^{4}
47D4D_{4} 1−18T+158T2−18pT3+p2T4 1 - 18 T + 158 T^{2} - 18 p T^{3} + p^{2} T^{4}
53D4D_{4} 1−11T+132T2−11pT3+p2T4 1 - 11 T + 132 T^{2} - 11 p T^{3} + p^{2} T^{4}
59C2C_2 (1−12T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}
61D4D_{4} 1−10T+130T2−10pT3+p2T4 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4}
67D4D_{4} 1+2T−18T2+2pT3+p2T4 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+14T+174T2+14pT3+p2T4 1 + 14 T + 174 T^{2} + 14 p T^{3} + p^{2} T^{4}
73D4D_{4} 1−9T+128T2−9pT3+p2T4 1 - 9 T + 128 T^{2} - 9 p T^{3} + p^{2} T^{4}
79C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
83D4D_{4} 1−T+162T2−pT3+p2T4 1 - T + 162 T^{2} - p T^{3} + p^{2} T^{4}
89C4C_4 1−8T+126T2−8pT3+p2T4 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+T+156T2+pT3+p2T4 1 + T + 156 T^{2} + 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

−9.313222484854293335946781178968, −8.768828903644330276023609874055, −8.456027232849861413611735095652, −8.223774013429579429353856889744, −7.67028272507488677244890318114, −7.18347387849391872063882335077, −7.14973796147253365657969554588, −6.40092069596639209662367063643, −6.08519516462701222910325816287, −5.73218190932137707230511023857, −5.35367422485844545339458319311, −5.35043315509478588001394626005, −4.29007800986332889679917777341, −4.02152142549637783535081537172, −3.47490252229220580230117733943, −2.71097211105444074065876313499, −2.62268070207715116817454999560, −1.83965249723161771003265248928, −0.69959274062441458301279973544, −0.64227985210694428018150375061, 0.64227985210694428018150375061, 0.69959274062441458301279973544, 1.83965249723161771003265248928, 2.62268070207715116817454999560, 2.71097211105444074065876313499, 3.47490252229220580230117733943, 4.02152142549637783535081537172, 4.29007800986332889679917777341, 5.35043315509478588001394626005, 5.35367422485844545339458319311, 5.73218190932137707230511023857, 6.08519516462701222910325816287, 6.40092069596639209662367063643, 7.14973796147253365657969554588, 7.18347387849391872063882335077, 7.67028272507488677244890318114, 8.223774013429579429353856889744, 8.456027232849861413611735095652, 8.768828903644330276023609874055, 9.313222484854293335946781178968

Graph of the ZZ-function along the critical line