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Properties

Label 2-327360-1.1-c1-0-80
Degree 22
Conductor 327360327360
Sign 11
Analytic cond. 2613.982613.98
Root an. cond. 51.127151.1271
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  − 3-s − 5-s + 4·7-s + 9-s + 11-s + 6·13-s + 15-s + 6·17-s − 4·21-s − 8·23-s + 25-s − 27-s − 2·29-s + 31-s − 33-s − 4·35-s + 2·37-s − 6·39-s − 6·41-s − 12·43-s − 45-s + 9·49-s − 6·51-s + 2·53-s − 55-s + 4·59-s − 2·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s + 1.66·13-s + 0.258·15-s + 1.45·17-s − 0.872·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.179·31-s − 0.174·33-s − 0.676·35-s + 0.328·37-s − 0.960·39-s − 0.937·41-s − 1.82·43-s − 0.149·45-s + 9/7·49-s − 0.840·51-s + 0.274·53-s − 0.134·55-s + 0.520·59-s − 0.256·61-s + ⋯

Functional equation

Λ(s)=(327360s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 327360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(327360s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 327360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 327360327360    =    26⋅3⋅5⋅11⋅312^{6} \cdot 3 \cdot 5 \cdot 11 \cdot 31
Sign: 11
Analytic conductor: 2613.982613.98
Root analytic conductor: 51.127151.1271
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 327360, ( :1/2), 1)(2,\ 327360,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.1953656343.195365634
L(12)L(\frac12) ≈\approx 3.1953656343.195365634
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}
ppFp(T)F_p(T)
bad2 1 1
3 1+T 1 + T
5 1+T 1 + T
11 1−T 1 - T
31 1−T 1 - T
good7 1−4T+pT2 1 - 4 T + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+12T+pT2 1 + 12 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1−18T+pT2 1 - 18 T + p 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

−12.43806075853047, −11.89389435027809, −11.67548662684109, −11.46425458672876, −10.80268553801978, −10.46096213401947, −10.01830570418203, −9.490853375601214, −8.723832665951349, −8.283433773697076, −8.188217377484606, −7.532986458784682, −7.177403415515383, −6.369829674146879, −6.019735702952666, −5.626353958618101, −4.958988614129694, −4.669463668449082, −3.943481849755083, −3.617772428028976, −3.139852892026403, −1.953668572281335, −1.748009644155581, −1.099984244012883, −0.5458753683376639, 0.5458753683376639, 1.099984244012883, 1.748009644155581, 1.953668572281335, 3.139852892026403, 3.617772428028976, 3.943481849755083, 4.669463668449082, 4.958988614129694, 5.626353958618101, 6.019735702952666, 6.369829674146879, 7.177403415515383, 7.532986458784682, 8.188217377484606, 8.283433773697076, 8.723832665951349, 9.490853375601214, 10.01830570418203, 10.46096213401947, 10.80268553801978, 11.46425458672876, 11.67548662684109, 11.89389435027809, 12.43806075853047

Graph of the ZZ-function along the critical line