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Properties

Label 2-336-1.1-c7-0-6
Degree 22
Conductor 336336
Sign 11
Analytic cond. 104.961104.961
Root an. cond. 10.245010.2450
Motivic weight 77
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 27·3-s − 18·5-s − 343·7-s + 729·9-s − 8.17e3·11-s − 1.42e4·13-s − 486·15-s − 2.14e4·17-s + 5.88e3·19-s − 9.26e3·21-s + 9.87e4·23-s − 7.78e4·25-s + 1.96e4·27-s + 1.65e5·29-s + 2.41e5·31-s − 2.20e5·33-s + 6.17e3·35-s + 1.85e5·37-s − 3.84e5·39-s + 5.96e4·41-s + 8.09e5·43-s − 1.31e4·45-s − 9.42e5·47-s + 1.17e5·49-s − 5.79e5·51-s + 2.26e5·53-s + 1.47e5·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.0643·5-s − 0.377·7-s + 1/3·9-s − 1.85·11-s − 1.79·13-s − 0.0371·15-s − 1.05·17-s + 0.196·19-s − 0.218·21-s + 1.69·23-s − 0.995·25-s + 0.192·27-s + 1.25·29-s + 1.45·31-s − 1.06·33-s + 0.0243·35-s + 0.601·37-s − 1.03·39-s + 0.135·41-s + 1.55·43-s − 0.0214·45-s − 1.32·47-s + 1/7·49-s − 0.611·51-s + 0.208·53-s + 0.119·55-s + ⋯

Functional equation

Λ(s)=(336s/2 ΓC(s) L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(336s/2 ΓC(s+7/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 336336    =    24⋅3⋅72^{4} \cdot 3 \cdot 7
Sign: 11
Analytic conductor: 104.961104.961
Root analytic conductor: 10.245010.2450
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 336, ( :7/2), 1)(2,\ 336,\ (\ :7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 1.5746087911.574608791
L(12)L(\frac12) ≈\approx 1.5746087911.574608791
L(92)L(\frac{9}{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}
ppFp(T)F_p(T)
bad2 1 1
3 1−p3T 1 - p^{3} T
7 1+p3T 1 + p^{3} T
good5 1+18T+p7T2 1 + 18 T + p^{7} T^{2}
11 1+8172T+p7T2 1 + 8172 T + p^{7} T^{2}
13 1+14242T+p7T2 1 + 14242 T + p^{7} T^{2}
17 1+21462T+p7T2 1 + 21462 T + p^{7} T^{2}
19 1−5884T+p7T2 1 - 5884 T + p^{7} T^{2}
23 1−98784T+p7T2 1 - 98784 T + p^{7} T^{2}
29 1−165174T+p7T2 1 - 165174 T + p^{7} T^{2}
31 1−241312T+p7T2 1 - 241312 T + p^{7} T^{2}
37 1−185438T+p7T2 1 - 185438 T + p^{7} T^{2}
41 1−59682T+p7T2 1 - 59682 T + p^{7} T^{2}
43 1−809308T+p7T2 1 - 809308 T + p^{7} T^{2}
47 1+942096T+p7T2 1 + 942096 T + p^{7} T^{2}
53 1−226398T+p7T2 1 - 226398 T + p^{7} T^{2}
59 1−2205732T+p7T2 1 - 2205732 T + p^{7} T^{2}
61 1+1156690T+p7T2 1 + 1156690 T + p^{7} T^{2}
67 1−3740404T+p7T2 1 - 3740404 T + p^{7} T^{2}
71 1−2593296T+p7T2 1 - 2593296 T + p^{7} T^{2}
73 1+1038742T+p7T2 1 + 1038742 T + p^{7} T^{2}
79 1+2280032T+p7T2 1 + 2280032 T + p^{7} T^{2}
83 1−283404T+p7T2 1 - 283404 T + p^{7} T^{2}
89 1+5227230T+p7T2 1 + 5227230 T + p^{7} T^{2}
97 1−6168770T+p7T2 1 - 6168770 T + p^{7} T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.15169264930751325884542090445, −9.560751493136091847202281625442, −8.373787111207332661564329029926, −7.58918404875320912701682763041, −6.73170219722989646104878986482, −5.24539060509774471326636344483, −4.49131744632304455943523559193, −2.80749075884772852403606027762, −2.45098324817472424885529743416, −0.54833022780874073047632440046, 0.54833022780874073047632440046, 2.45098324817472424885529743416, 2.80749075884772852403606027762, 4.49131744632304455943523559193, 5.24539060509774471326636344483, 6.73170219722989646104878986482, 7.58918404875320912701682763041, 8.373787111207332661564329029926, 9.560751493136091847202281625442, 10.15169264930751325884542090445

Graph of the ZZ-function along the critical line