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Properties

Label 2-3800-152.11-c0-0-2
Degree 22
Conductor 38003800
Sign −0.211−0.977i-0.211 - 0.977i
Analytic cond. 1.896441.89644
Root an. cond. 1.377111.37711
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + i·7-s + 0.999i·8-s + (0.5 + 0.866i)9-s − 11-s + (1.73 − i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + 0.999i·18-s + (0.5 − 0.866i)19-s + (−0.866 − 0.5i)22-s + (−0.866 + 0.5i)23-s + 1.99·26-s + (−0.866 + 0.499i)28-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + i·7-s + 0.999i·8-s + (0.5 + 0.866i)9-s − 11-s + (1.73 − i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + 0.999i·18-s + (0.5 − 0.866i)19-s + (−0.866 − 0.5i)22-s + (−0.866 + 0.5i)23-s + 1.99·26-s + (−0.866 + 0.499i)28-s + ⋯

Functional equation

Λ(s)=(3800s/2 ΓC(s) L(s)=((−0.211−0.977i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3800s/2 ΓC(s) L(s)=((−0.211−0.977i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: −0.211−0.977i-0.211 - 0.977i
Analytic conductor: 1.896441.89644
Root analytic conductor: 1.377111.37711
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3800(3051,⋅)\chi_{3800} (3051, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3800, ( :0), −0.211−0.977i)(2,\ 3800,\ (\ :0),\ -0.211 - 0.977i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.2378591852.237859185
L(12)L(\frac12) ≈\approx 2.2378591852.237859185
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+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
5 1 1
19 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good3 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
7 1−iT−T2 1 - iT - T^{2}
11 1+T+T2 1 + T + T^{2}
13 1+(−1.73+i)T+(0.5−0.866i)T2 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2}
17 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
23 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
29 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
31 1−T2 1 - T^{2}
37 1−iT−T2 1 - iT - T^{2}
41 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
43 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
47 1+(1.73−i)T+(0.5−0.866i)T2 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2}
53 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
59 1+(−1+1.73i)T+(−0.5−0.866i)T2 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2}
61 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
67 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
71 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
73 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1+T2 1 + T^{2}
89 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
97 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)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

−8.428761861482699882193894562517, −8.154490171165269135437796560112, −7.46267607951210365773007206612, −6.47632951692374417876519499215, −5.73851315685485723796054475412, −5.28227786592011338161968098677, −4.52470547421764747266098938803, −3.41227592284891481411068724519, −2.76902908826341043465858410111, −1.76748742974104671036151173035, 1.01708103101501137684679887016, 1.90659575792252689265234237592, 3.23533613497745801465096609311, 3.91660813955462596166632400786, 4.32608151548375150930967587697, 5.45793327016459922882346401632, 6.23028220204478854160434697400, 6.74940506641070607224091097162, 7.58508436093900464672725234231, 8.447204416373063899717721441716

Graph of the ZZ-function along the critical line