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Properties

Label 8-2400e4-1.1-c2e4-0-1
Degree 88
Conductor 3.318×10133.318\times 10^{13}
Sign 11
Analytic cond. 1.82887×1071.82887\times 10^{7}
Root an. cond. 8.086738.08673
Motivic weight 22
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  + 6·9-s + 32·11-s + 8·17-s − 32·19-s + 40·41-s + 32·43-s + 76·49-s + 128·59-s − 256·67-s − 200·73-s + 27·81-s + 160·83-s − 200·89-s − 56·97-s + 192·99-s − 320·107-s + 344·113-s + 156·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 48·153-s + 157-s + 163-s + ⋯
L(s)  = 1  + 2/3·9-s + 2.90·11-s + 8/17·17-s − 1.68·19-s + 0.975·41-s + 0.744·43-s + 1.55·49-s + 2.16·59-s − 3.82·67-s − 2.73·73-s + 1/3·81-s + 1.92·83-s − 2.24·89-s − 0.577·97-s + 1.93·99-s − 2.99·107-s + 3.04·113-s + 1.28·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.313·153-s + 0.00636·157-s + 0.00613·163-s + ⋯

Functional equation

Λ(s)=((220⋅34⋅58)s/2 ΓC(s)4 L(s)=( Λ(3−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}
Λ(s)=((220⋅34⋅58)s/2 ΓC(s+1)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 220⋅34⋅582^{20} \cdot 3^{4} \cdot 5^{8}
Sign: 11
Analytic conductor: 1.82887×1071.82887\times 10^{7}
Root analytic conductor: 8.086738.08673
Motivic weight: 22
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 220⋅34⋅58, ( :1,1,1,1), 1)(8,\ 2^{20} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 2.3561065512.356106551
L(12)L(\frac12) ≈\approx 2.3561065512.356106551
L(2)L(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
3C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
5 1 1
good7C22≀C2C_2^2 \wr C_2 1−76T2+3174T4−76p4T6+p8T8 1 - 76 T^{2} + 3174 T^{4} - 76 p^{4} T^{6} + p^{8} T^{8}
11C2C_2 (1−8T+p2T2)4 ( 1 - 8 T + p^{2} T^{2} )^{4}
13C22≀C2C_2^2 \wr C_2 1−292T2+75366T4−292p4T6+p8T8 1 - 292 T^{2} + 75366 T^{4} - 292 p^{4} T^{6} + p^{8} T^{8}
17D4D_{4} (1−4T+390T2−4p2T3+p4T4)2 ( 1 - 4 T + 390 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2}
19D4D_{4} (1+16T+738T2+16p2T3+p4T4)2 ( 1 + 16 T + 738 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2}
23C22≀C2C_2^2 \wr C_2 1−1636T2+1179654T4−1636p4T6+p8T8 1 - 1636 T^{2} + 1179654 T^{4} - 1636 p^{4} T^{6} + p^{8} T^{8}
29C22≀C2C_2^2 \wr C_2 1−1756T2+1539558T4−1756p4T6+p8T8 1 - 1756 T^{2} + 1539558 T^{4} - 1756 p^{4} T^{6} + p^{8} T^{8}
31C22≀C2C_2^2 \wr C_2 1−460T2−683610T4−460p4T6+p8T8 1 - 460 T^{2} - 683610 T^{4} - 460 p^{4} T^{6} + p^{8} T^{8}
37C22≀C2C_2^2 \wr C_2 1−4612T2+8955366T4−4612p4T6+p8T8 1 - 4612 T^{2} + 8955366 T^{4} - 4612 p^{4} T^{6} + p^{8} T^{8}
41D4D_{4} (1−20T+1734T2−20p2T3+p4T4)2 ( 1 - 20 T + 1734 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2}
43D4D_{4} (1−16T+3330T2−16p2T3+p4T4)2 ( 1 - 16 T + 3330 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2}
47C22≀C2C_2^2 \wr C_2 1−5284T2+13593798T4−5284p4T6+p8T8 1 - 5284 T^{2} + 13593798 T^{4} - 5284 p^{4} T^{6} + p^{8} T^{8}
53C22≀C2C_2^2 \wr C_2 1−9436T2+38033574T4−9436p4T6+p8T8 1 - 9436 T^{2} + 38033574 T^{4} - 9436 p^{4} T^{6} + p^{8} T^{8}
59D4D_{4} (1−64T+7554T2−64p2T3+p4T4)2 ( 1 - 64 T + 7554 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2}
61C22≀C2C_2^2 \wr C_2 1−11332T2+56649510T4−11332p4T6+p8T8 1 - 11332 T^{2} + 56649510 T^{4} - 11332 p^{4} T^{6} + p^{8} T^{8}
67D4D_{4} (1+128T+12642T2+128p2T3+p4T4)2 ( 1 + 128 T + 12642 T^{2} + 128 p^{2} T^{3} + p^{4} T^{4} )^{2}
71C22≀C2C_2^2 \wr C_2 1−11236T2+75307398T4−11236p4T6+p8T8 1 - 11236 T^{2} + 75307398 T^{4} - 11236 p^{4} T^{6} + p^{8} T^{8}
73D4D_{4} (1+100T+10086T2+100p2T3+p4T4)2 ( 1 + 100 T + 10086 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2}
79C22≀C2C_2^2 \wr C_2 1−17548T2+151931046T4−17548p4T6+p8T8 1 - 17548 T^{2} + 151931046 T^{4} - 17548 p^{4} T^{6} + p^{8} T^{8}
83D4D_{4} (1−80T+14610T2−80p2T3+p4T4)2 ( 1 - 80 T + 14610 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{2}
89D4D_{4} (1+100T+11430T2+100p2T3+p4T4)2 ( 1 + 100 T + 11430 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2}
97D4D_{4} (1+28T+12102T2+28p2T3+p4T4)2 ( 1 + 28 T + 12102 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−6.30837432617801267022212997253, −5.99852341962762696348607485779, −5.97453157257984598381081181868, −5.43038858108407311699709910724, −5.40154869760803525790748042018, −5.16790227980571358819034513640, −5.02697131928056970415322728439, −4.36640152925299006331163329614, −4.28995315262312354728827752575, −4.27681553971705068375822292030, −4.14201753226866515679781318300, −3.99386139666729341941121198164, −3.71084394370455828260012367088, −3.40557016520608517243966409130, −3.12864895466725836463511195270, −2.90004736955513344832554781072, −2.46382369906127906732552316927, −2.29250095185819982617812541981, −2.27909898829777739531796402702, −1.46692307426145807498682876334, −1.45555949584384425579074327990, −1.36812250922288461757431683009, −1.10424320769343015670376940787, −0.60679995983731705730875909482, −0.16706220649453262561616381310, 0.16706220649453262561616381310, 0.60679995983731705730875909482, 1.10424320769343015670376940787, 1.36812250922288461757431683009, 1.45555949584384425579074327990, 1.46692307426145807498682876334, 2.27909898829777739531796402702, 2.29250095185819982617812541981, 2.46382369906127906732552316927, 2.90004736955513344832554781072, 3.12864895466725836463511195270, 3.40557016520608517243966409130, 3.71084394370455828260012367088, 3.99386139666729341941121198164, 4.14201753226866515679781318300, 4.27681553971705068375822292030, 4.28995315262312354728827752575, 4.36640152925299006331163329614, 5.02697131928056970415322728439, 5.16790227980571358819034513640, 5.40154869760803525790748042018, 5.43038858108407311699709910724, 5.97453157257984598381081181868, 5.99852341962762696348607485779, 6.30837432617801267022212997253

Graph of the ZZ-function along the critical line