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Properties

Label 2-4002-1.1-c1-0-36
Degree 22
Conductor 40024002
Sign 11
Analytic cond. 31.956131.9561
Root an. cond. 5.652975.65297
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  + 2-s − 3-s + 4-s + 2·5-s − 6-s + 4·7-s + 8-s + 9-s + 2·10-s − 12-s − 2·13-s + 4·14-s − 2·15-s + 16-s − 6·17-s + 18-s + 4·19-s + 2·20-s − 4·21-s + 23-s − 24-s − 25-s − 2·26-s − 27-s + 4·28-s + 29-s − 2·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s − 0.554·13-s + 1.06·14-s − 0.516·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.447·20-s − 0.872·21-s + 0.208·23-s − 0.204·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.755·28-s + 0.185·29-s − 0.365·30-s + ⋯

Functional equation

Λ(s)=(4002s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4002s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40024002    =    2⋅3⋅23⋅292 \cdot 3 \cdot 23 \cdot 29
Sign: 11
Analytic conductor: 31.956131.9561
Root analytic conductor: 5.652975.65297
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4002, ( :1/2), 1)(2,\ 4002,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.6563870653.656387065
L(12)L(\frac12) ≈\approx 3.6563870653.656387065
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−T 1 - T
3 1+T 1 + T
23 1−T 1 - T
29 1−T 1 - T
good5 1−2T+pT2 1 - 2 T + p T^{2}
7 1−4T+pT2 1 - 4 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1+14T+pT2 1 + 14 T + p T^{2}
67 1−16T+pT2 1 - 16 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1+16T+pT2 1 + 16 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−10T+pT2 1 - 10 T + p T^{2}
97 1−6T+pT2 1 - 6 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

−8.263585704963024975830727045832, −7.63156864718409983699746942194, −6.80409326667564881991132180521, −6.04219302925733683856174147153, −5.44674612443065479179177428541, −4.60748063880942386605687322558, −4.36934955938838586718487397549, −2.76749490203986529835135686893, −2.05401603112340154266513492577, −1.09378025244341600177404013908, 1.09378025244341600177404013908, 2.05401603112340154266513492577, 2.76749490203986529835135686893, 4.36934955938838586718487397549, 4.60748063880942386605687322558, 5.44674612443065479179177428541, 6.04219302925733683856174147153, 6.80409326667564881991132180521, 7.63156864718409983699746942194, 8.263585704963024975830727045832

Graph of the ZZ-function along the critical line