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Properties

Label 4-704e2-1.1-c1e2-0-4
Degree 44
Conductor 495616495616
Sign 11
Analytic cond. 31.600931.6009
Root an. cond. 2.370962.37096
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 2·5-s + 7·9-s + 4·11-s − 8·15-s + 8·23-s − 3·25-s + 4·27-s + 16·33-s + 6·37-s − 14·45-s + 6·49-s − 4·53-s − 8·55-s + 12·59-s + 4·67-s + 32·69-s + 24·71-s − 12·75-s − 8·81-s − 26·89-s + 6·97-s + 28·99-s − 16·103-s + 24·111-s − 10·113-s − 16·115-s + ⋯
L(s)  = 1  + 2.30·3-s − 0.894·5-s + 7/3·9-s + 1.20·11-s − 2.06·15-s + 1.66·23-s − 3/5·25-s + 0.769·27-s + 2.78·33-s + 0.986·37-s − 2.08·45-s + 6/7·49-s − 0.549·53-s − 1.07·55-s + 1.56·59-s + 0.488·67-s + 3.85·69-s + 2.84·71-s − 1.38·75-s − 8/9·81-s − 2.75·89-s + 0.609·97-s + 2.81·99-s − 1.57·103-s + 2.27·111-s − 0.940·113-s − 1.49·115-s + ⋯

Functional equation

Λ(s)=(495616s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(495616s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 495616495616    =    212⋅1122^{12} \cdot 11^{2}
Sign: 11
Analytic conductor: 31.600931.6009
Root analytic conductor: 2.370962.37096
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 495616, ( :1/2,1/2), 1)(4,\ 495616,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.0403424534.040342453
L(12)L(\frac12) ≈\approx 4.0403424534.040342453
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)
bad2 1 1
11C2C_2 1−4T+pT2 1 - 4 T + p T^{2}
good3C2C_2×\timesC2C_2 (1−pT+pT2)(1−T+pT2) ( 1 - p T + p T^{2} )( 1 - T + p T^{2} )
5C2C_2×\timesC2C_2 (1−T+pT2)(1+3T+pT2) ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} )
7C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
13C22C_2^2 1−6T2+p2T4 1 - 6 T^{2} + p^{2} T^{4}
17C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
19C22C_2^2 1+18T2+p2T4 1 + 18 T^{2} + p^{2} T^{4}
23C2C_2×\timesC2C_2 (1−7T+pT2)(1−T+pT2) ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} )
29C22C_2^2 1+6T2+p2T4 1 + 6 T^{2} + p^{2} T^{4}
31C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
37C2C_2×\timesC2C_2 (1−9T+pT2)(1+3T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} )
41C22C_2^2 1−30T2+p2T4 1 - 30 T^{2} + p^{2} T^{4}
43C22C_2^2 1+54T2+p2T4 1 + 54 T^{2} + p^{2} T^{4}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C2C_2×\timesC2C_2 (1−6T+pT2)(1+10T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} )
59C2C_2×\timesC2C_2 (1−7T+pT2)(1−5T+pT2) ( 1 - 7 T + p T^{2} )( 1 - 5 T + p T^{2} )
61C22C_2^2 1−58T2+p2T4 1 - 58 T^{2} + p^{2} T^{4}
67C2C_2×\timesC2C_2 (1−9T+pT2)(1+5T+pT2) ( 1 - 9 T + p T^{2} )( 1 + 5 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−15T+pT2)(1−9T+pT2) ( 1 - 15 T + p T^{2} )( 1 - 9 T + p T^{2} )
73C22C_2^2 1+66T2+p2T4 1 + 66 T^{2} + p^{2} T^{4}
79C22C_2^2 1−114T2+p2T4 1 - 114 T^{2} + p^{2} T^{4}
83C22C_2^2 1+86T2+p2T4 1 + 86 T^{2} + p^{2} T^{4}
89C2C_2×\timesC2C_2 (1+9T+pT2)(1+17T+pT2) ( 1 + 9 T + p T^{2} )( 1 + 17 T + p T^{2} )
97C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
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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

−8.527834757835193726747029498818, −8.130905320346842230293799253614, −7.80880047675026696994038925633, −7.38343675003183133018650001814, −6.75675677658737613905670147750, −6.62082714324451009339742326913, −5.66163434916591204950710996791, −5.16707365592353737946128191517, −4.30971861340138374132199477742, −3.95501966037150973513946344449, −3.64483685966352076698862940684, −3.01131321872163071924120162853, −2.60939963678351254594850012675, −1.92256304918733355678144886727, −0.989457762310743048355362377137, 0.989457762310743048355362377137, 1.92256304918733355678144886727, 2.60939963678351254594850012675, 3.01131321872163071924120162853, 3.64483685966352076698862940684, 3.95501966037150973513946344449, 4.30971861340138374132199477742, 5.16707365592353737946128191517, 5.66163434916591204950710996791, 6.62082714324451009339742326913, 6.75675677658737613905670147750, 7.38343675003183133018650001814, 7.80880047675026696994038925633, 8.130905320346842230293799253614, 8.527834757835193726747029498818

Graph of the ZZ-function along the critical line