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Properties

Label 2-4002-1.1-c1-0-19
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 2.5824310802.582431080
L(12)L(\frac12) ≈\approx 2.5824310802.582431080
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+pT2 1 + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1−2T+pT2 1 - 2 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+10T+pT2 1 + 10 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1−8T+pT2 1 - 8 T + p T^{2}
89 1−14T+pT2 1 - 14 T + p T^{2}
97 1−10T+pT2 1 - 10 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

−8.117923352354560842576493810952, −7.72241330075962692237819063696, −7.01333323419057346375191818255, −6.24812917471354955721910234661, −5.46356916712438031826540105298, −4.50162287690062263138037116263, −3.55784116503777040358801542907, −3.29995376479477247004032977433, −2.38745920662164929006825210076, −0.792012704452018534737806150565, 0.792012704452018534737806150565, 2.38745920662164929006825210076, 3.29995376479477247004032977433, 3.55784116503777040358801542907, 4.50162287690062263138037116263, 5.46356916712438031826540105298, 6.24812917471354955721910234661, 7.01333323419057346375191818255, 7.72241330075962692237819063696, 8.117923352354560842576493810952

Graph of the ZZ-function along the critical line