↑

Properties

Label 2-1872-13.4-c1-0-6
Degree 22
Conductor 18721872
Sign 0.265−0.964i0.265 - 0.964i
Analytic cond. 14.947914.9479
Root an. cond. 3.866263.86626
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.73i·5-s + (−2.36 + 1.36i)7-s + (1.09 + 0.633i)11-s + (−2.59 + 2.5i)13-s + (2.86 + 4.96i)17-s + (−4.09 + 2.36i)19-s + (2.09 − 3.63i)23-s − 8.92·25-s + (−2.23 + 3.86i)29-s − 1.46i·31-s + (5.09 + 8.83i)35-s + (3.06 + 1.76i)37-s + (−8.13 − 4.69i)41-s + (4.83 + 8.36i)43-s − 2.19i·47-s + ⋯
L(s)  = 1  − 1.66i·5-s + (−0.894 + 0.516i)7-s + (0.331 + 0.191i)11-s + (−0.720 + 0.693i)13-s + (0.695 + 1.20i)17-s + (−0.940 + 0.542i)19-s + (0.437 − 0.757i)23-s − 1.78·25-s + (−0.414 + 0.717i)29-s − 0.262i·31-s + (0.861 + 1.49i)35-s + (0.503 + 0.290i)37-s + (−1.27 − 0.733i)41-s + (0.736 + 1.27i)43-s − 0.320i·47-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=((0.265−0.964i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s+1/2) L(s)=((0.265−0.964i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: 0.265−0.964i0.265 - 0.964i
Analytic conductor: 14.947914.9479
Root analytic conductor: 3.866263.86626
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1872(433,⋅)\chi_{1872} (433, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1872, ( :1/2), 0.265−0.964i)(2,\ 1872,\ (\ :1/2),\ 0.265 - 0.964i)

Particular Values

L(1)L(1) ≈\approx 0.89002750080.8900275008
L(12)L(\frac12) ≈\approx 0.89002750080.8900275008
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
13 1+(2.59−2.5i)T 1 + (2.59 - 2.5i)T
good5 1+3.73iT−5T2 1 + 3.73iT - 5T^{2}
7 1+(2.36−1.36i)T+(3.5−6.06i)T2 1 + (2.36 - 1.36i)T + (3.5 - 6.06i)T^{2}
11 1+(−1.09−0.633i)T+(5.5+9.52i)T2 1 + (-1.09 - 0.633i)T + (5.5 + 9.52i)T^{2}
17 1+(−2.86−4.96i)T+(−8.5+14.7i)T2 1 + (-2.86 - 4.96i)T + (-8.5 + 14.7i)T^{2}
19 1+(4.09−2.36i)T+(9.5−16.4i)T2 1 + (4.09 - 2.36i)T + (9.5 - 16.4i)T^{2}
23 1+(−2.09+3.63i)T+(−11.5−19.9i)T2 1 + (-2.09 + 3.63i)T + (-11.5 - 19.9i)T^{2}
29 1+(2.23−3.86i)T+(−14.5−25.1i)T2 1 + (2.23 - 3.86i)T + (-14.5 - 25.1i)T^{2}
31 1+1.46iT−31T2 1 + 1.46iT - 31T^{2}
37 1+(−3.06−1.76i)T+(18.5+32.0i)T2 1 + (-3.06 - 1.76i)T + (18.5 + 32.0i)T^{2}
41 1+(8.13+4.69i)T+(20.5+35.5i)T2 1 + (8.13 + 4.69i)T + (20.5 + 35.5i)T^{2}
43 1+(−4.83−8.36i)T+(−21.5+37.2i)T2 1 + (-4.83 - 8.36i)T + (-21.5 + 37.2i)T^{2}
47 1+2.19iT−47T2 1 + 2.19iT - 47T^{2}
53 1−6.46T+53T2 1 - 6.46T + 53T^{2}
59 1+(6.92−4i)T+(29.5−51.0i)T2 1 + (6.92 - 4i)T + (29.5 - 51.0i)T^{2}
61 1+(−4.59−7.96i)T+(−30.5+52.8i)T2 1 + (-4.59 - 7.96i)T + (-30.5 + 52.8i)T^{2}
67 1+(−11.3−6.56i)T+(33.5+58.0i)T2 1 + (-11.3 - 6.56i)T + (33.5 + 58.0i)T^{2}
71 1+(−4.09+2.36i)T+(35.5−61.4i)T2 1 + (-4.09 + 2.36i)T + (35.5 - 61.4i)T^{2}
73 1−6.26iT−73T2 1 - 6.26iT - 73T^{2}
79 1−2.53T+79T2 1 - 2.53T + 79T^{2}
83 1+0.196iT−83T2 1 + 0.196iT - 83T^{2}
89 1+(−8.19−4.73i)T+(44.5+77.0i)T2 1 + (-8.19 - 4.73i)T + (44.5 + 77.0i)T^{2}
97 1+(5.19−3i)T+(48.5−84.0i)T2 1 + (5.19 - 3i)T + (48.5 - 84.0i)T^{2}
show more
show less
   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

−9.346186156061080273315979023549, −8.643283884392320091544962346986, −8.126711545095316445409112120493, −6.94384481378956860495678577596, −6.10319589174468896248413298343, −5.36713891312074962798839302427, −4.45967987150735224544544887405, −3.75712707449025414375243915440, −2.31863657989459619960299699898, −1.23330870726592085771285865537, 0.34156125650097746099352801886, 2.34667998785262292360818744669, 3.14276534913239257187332781737, 3.73781223880333018593921508335, 5.07717583428523280393702250212, 6.08173485506847640583162557992, 6.86332458161479582813863519860, 7.23549015905815311025147182605, 8.065915433573776297068582115981, 9.421111605227269641797062503631

Graph of the ZZ-function along the critical line