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Properties

Label 2-40e2-40.13-c0-0-2
Degree 22
Conductor 16001600
Sign 0.945−0.326i0.945 - 0.326i
Analytic cond. 0.7985040.798504
Root an. cond. 0.8935900.893590
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 1.22i)3-s − 1.99i·9-s − i·11-s + (1.22 − 1.22i)17-s + 19-s + (1.22 + 1.22i)27-s + (1.22 + 1.22i)33-s − 41-s + i·49-s + 2.99i·51-s + (−1.22 + 1.22i)57-s + 2·59-s + (−1.22 − 1.22i)67-s + (1.22 + 1.22i)73-s − 0.999·81-s + ⋯
L(s)  = 1  + (−1.22 + 1.22i)3-s − 1.99i·9-s − i·11-s + (1.22 − 1.22i)17-s + 19-s + (1.22 + 1.22i)27-s + (1.22 + 1.22i)33-s − 41-s + i·49-s + 2.99i·51-s + (−1.22 + 1.22i)57-s + 2·59-s + (−1.22 − 1.22i)67-s + (1.22 + 1.22i)73-s − 0.999·81-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=((0.945−0.326i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s) L(s)=((0.945−0.326i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: 0.945−0.326i0.945 - 0.326i
Analytic conductor: 0.7985040.798504
Root analytic conductor: 0.8935900.893590
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1600(993,⋅)\chi_{1600} (993, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1600, ( :0), 0.945−0.326i)(2,\ 1600,\ (\ :0),\ 0.945 - 0.326i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.74198617900.7419861790
L(12)L(\frac12) ≈\approx 0.74198617900.7419861790
L(1)L(1) 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
5 1 1
good3 1+(1.22−1.22i)T−iT2 1 + (1.22 - 1.22i)T - iT^{2}
7 1−iT2 1 - iT^{2}
11 1+iT−T2 1 + iT - T^{2}
13 1−iT2 1 - iT^{2}
17 1+(−1.22+1.22i)T−iT2 1 + (-1.22 + 1.22i)T - iT^{2}
19 1−T+T2 1 - T + T^{2}
23 1+iT2 1 + iT^{2}
29 1+T2 1 + T^{2}
31 1+T2 1 + T^{2}
37 1+iT2 1 + iT^{2}
41 1+T+T2 1 + T + T^{2}
43 1−iT2 1 - iT^{2}
47 1−iT2 1 - iT^{2}
53 1−iT2 1 - iT^{2}
59 1−2T+T2 1 - 2T + T^{2}
61 1−T2 1 - T^{2}
67 1+(1.22+1.22i)T+iT2 1 + (1.22 + 1.22i)T + iT^{2}
71 1+T2 1 + T^{2}
73 1+(−1.22−1.22i)T+iT2 1 + (-1.22 - 1.22i)T + iT^{2}
79 1−T2 1 - T^{2}
83 1+(−1.22+1.22i)T−iT2 1 + (-1.22 + 1.22i)T - iT^{2}
89 1−iT−T2 1 - iT - T^{2}
97 1−iT2 1 - iT^{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

−9.714720786236130314317281861911, −9.211079371241800975188443617768, −8.108862813009212543179465396233, −7.09959911852944383639502627036, −6.11209615441944494090003650544, −5.40027115487692109183849460081, −4.92850778866561085010260028843, −3.76692036608279088586439958730, −3.04929714164307614392700433941, −0.854792395949630825824052265300, 1.17002212834116417923128429689, 2.09494165064015195096684179690, 3.61498629578203627218172465436, 4.93844714229888410008396902722, 5.56915779428869497575206393200, 6.35404777405152463038642474250, 7.14669936269317272762810026517, 7.67641948739512780711892624946, 8.518131259915099924681028763603, 9.852846750045062303110477711126

Graph of the ZZ-function along the critical line