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Properties

Label 4-1408e2-1.1-c1e2-0-2
Degree 44
Conductor 19824641982464
Sign 11
Analytic cond. 126.403126.403
Root an. cond. 3.353043.35304
Motivic weight 11
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  + 4·3-s − 4·5-s + 6·9-s − 2·11-s − 16·15-s − 8·23-s + 2·25-s − 4·27-s − 8·33-s − 20·37-s − 24·45-s + 16·47-s + 2·49-s + 12·53-s + 8·55-s + 28·59-s + 20·67-s − 32·69-s − 24·71-s + 8·75-s − 37·81-s − 4·89-s − 4·97-s − 12·99-s + 8·103-s − 80·111-s + 4·113-s + ⋯
L(s)  = 1  + 2.30·3-s − 1.78·5-s + 2·9-s − 0.603·11-s − 4.13·15-s − 1.66·23-s + 2/5·25-s − 0.769·27-s − 1.39·33-s − 3.28·37-s − 3.57·45-s + 2.33·47-s + 2/7·49-s + 1.64·53-s + 1.07·55-s + 3.64·59-s + 2.44·67-s − 3.85·69-s − 2.84·71-s + 0.923·75-s − 4.11·81-s − 0.423·89-s − 0.406·97-s − 1.20·99-s + 0.788·103-s − 7.59·111-s + 0.376·113-s + ⋯

Functional equation

Λ(s)=(1982464s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1982464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1982464s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1982464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 19824641982464    =    214⋅1122^{14} \cdot 11^{2}
Sign: 11
Analytic conductor: 126.403126.403
Root analytic conductor: 3.353043.35304
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1982464, ( :1/2,1/2), 1)(4,\ 1982464,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6718182971.671818297
L(12)L(\frac12) ≈\approx 1.6718182971.671818297
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+2T+pT2 1 + 2 T + p T^{2}
good3C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
5C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
7C2C_2 (1−4T+pT2)(1+4T+pT2) ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )
13C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
17C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
19C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
23C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
29C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
31C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
37C2C_2 (1+10T+pT2)2 ( 1 + 10 T + p T^{2} )^{2}
41C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
43C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
47C2C_2 (1−8T+pT2)2 ( 1 - 8 T + p T^{2} )^{2}
53C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
59C2C_2 (1−14T+pT2)2 ( 1 - 14 T + p T^{2} )^{2}
61C2C_2 (1−2T+pT2)(1+2T+pT2) ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )
67C2C_2 (1−10T+pT2)2 ( 1 - 10 T + p T^{2} )^{2}
71C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
73C2C_2 (1−14T+pT2)(1+14T+pT2) ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )
79C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
83C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
89C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
97C2C_2 (1+2T+pT2)2 ( 1 + 2 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

−8.051397256544459587474063215822, −7.33738117792255002879146989083, −7.28624614008190284194684658876, −6.95203079036524006513160995367, −5.94748687441979739999249646343, −5.57971042236015603393981708123, −5.16542238567969836254694300670, −4.22499492266902913993361093129, −3.91058908136761919842529082491, −3.74391345354838938964523130661, −3.39209478584729487520505941993, −2.63262233065457230393893973638, −2.30676446591740616355718333113, −1.82152115990232001116828319877, −0.42520644735956268098029720198, 0.42520644735956268098029720198, 1.82152115990232001116828319877, 2.30676446591740616355718333113, 2.63262233065457230393893973638, 3.39209478584729487520505941993, 3.74391345354838938964523130661, 3.91058908136761919842529082491, 4.22499492266902913993361093129, 5.16542238567969836254694300670, 5.57971042236015603393981708123, 5.94748687441979739999249646343, 6.95203079036524006513160995367, 7.28624614008190284194684658876, 7.33738117792255002879146989083, 8.051397256544459587474063215822

Graph of the ZZ-function along the critical line