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Properties

Label 2-5040-21.20-c1-0-51
Degree 22
Conductor 50405040
Sign −0.836+0.547i-0.836 + 0.547i
Analytic cond. 40.244640.2446
Root an. cond. 6.343866.34386
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + (−0.0951 + 2.64i)7-s − 5.28i·11-s + 2.19i·13-s + 1.04·17-s − 6.43i·19-s − 7.47i·23-s + 25-s + 7.47i·29-s + 9.09i·31-s + (0.0951 − 2.64i)35-s − 0.855·37-s − 2.19·41-s + 0.954·43-s − 11.0·47-s + ⋯
L(s)  = 1  − 0.447·5-s + (−0.0359 + 0.999i)7-s − 1.59i·11-s + 0.607i·13-s + 0.253·17-s − 1.47i·19-s − 1.55i·23-s + 0.200·25-s + 1.38i·29-s + 1.63i·31-s + (0.0160 − 0.446i)35-s − 0.140·37-s − 0.342·41-s + 0.145·43-s − 1.60·47-s + ⋯

Functional equation

Λ(s)=(5040s/2 ΓC(s) L(s)=((−0.836+0.547i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(5040s/2 ΓC(s+1/2) L(s)=((−0.836+0.547i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 50405040    =    24⋅32⋅5⋅72^{4} \cdot 3^{2} \cdot 5 \cdot 7
Sign: −0.836+0.547i-0.836 + 0.547i
Analytic conductor: 40.244640.2446
Root analytic conductor: 6.343866.34386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ5040(881,⋅)\chi_{5040} (881, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 5040, ( :1/2), −0.836+0.547i)(2,\ 5040,\ (\ :1/2),\ -0.836 + 0.547i)

Particular Values

L(1)L(1) ≈\approx 0.44138961540.4413896154
L(12)L(\frac12) ≈\approx 0.44138961540.4413896154
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 1
5 1+T 1 + T
7 1+(0.0951−2.64i)T 1 + (0.0951 - 2.64i)T
good11 1+5.28iT−11T2 1 + 5.28iT - 11T^{2}
13 1−2.19iT−13T2 1 - 2.19iT - 13T^{2}
17 1−1.04T+17T2 1 - 1.04T + 17T^{2}
19 1+6.43iT−19T2 1 + 6.43iT - 19T^{2}
23 1+7.47iT−23T2 1 + 7.47iT - 23T^{2}
29 1−7.47iT−29T2 1 - 7.47iT - 29T^{2}
31 1−9.09iT−31T2 1 - 9.09iT - 31T^{2}
37 1+0.855T+37T2 1 + 0.855T + 37T^{2}
41 1+2.19T+41T2 1 + 2.19T + 41T^{2}
43 1−0.954T+43T2 1 - 0.954T + 43T^{2}
47 1+11.0T+47T2 1 + 11.0T + 47T^{2}
53 1+3.09iT−53T2 1 + 3.09iT - 53T^{2}
59 1−13.7T+59T2 1 - 13.7T + 59T^{2}
61 1−8.05iT−61T2 1 - 8.05iT - 61T^{2}
67 1+5.33T+67T2 1 + 5.33T + 67T^{2}
71 1−6.43iT−71T2 1 - 6.43iT - 71T^{2}
73 1−4.57iT−73T2 1 - 4.57iT - 73T^{2}
79 1+15.6T+79T2 1 + 15.6T + 79T^{2}
83 1+4.38T+83T2 1 + 4.38T + 83T^{2}
89 1+4.28T+89T2 1 + 4.28T + 89T^{2}
97 1+11.8iT−97T2 1 + 11.8iT - 97T^{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

−8.321488228754995380761214920814, −6.91296433261602343972345568310, −6.73587007114032601103835795234, −5.67010615748649237727142520222, −5.10080844863234680826946994031, −4.25685383281580170905843231439, −3.11524479789980278047727264052, −2.78419743312902339519311792915, −1.40504303783983409760650919522, −0.12319444385561396371680421981, 1.26720492413828452035832569673, 2.19266821540687892750539315141, 3.49324481666117236229477623632, 3.98370186592904489465328207127, 4.74018045702458173166755507005, 5.61928165390656916164256966551, 6.43480578340247466277690654954, 7.35803816104168685381480749773, 7.71996349946965925555746278875, 8.182217301881173607146699625663

Graph of the ZZ-function along the critical line