↑

Properties

Label 2-30e2-300.167-c0-0-0
Degree 22
Conductor 900900
Sign 0.356+0.934i0.356 + 0.934i
Analytic cond. 0.4491580.449158
Root an. cond. 0.6701920.670192
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.891 − 0.453i)2-s + (0.587 + 0.809i)4-s + (−0.987 + 0.156i)5-s + (−0.156 − 0.987i)8-s + (0.951 + 0.309i)10-s + (−0.896 − 1.76i)13-s + (−0.309 + 0.951i)16-s + (1.87 − 0.297i)17-s + (−0.707 − 0.707i)20-s + (0.951 − 0.309i)25-s + 1.97i·26-s + (1.44 − 1.04i)29-s + (0.707 − 0.707i)32-s + (−1.80 − 0.587i)34-s + (0.809 − 0.412i)37-s + ⋯
L(s)  = 1  + (−0.891 − 0.453i)2-s + (0.587 + 0.809i)4-s + (−0.987 + 0.156i)5-s + (−0.156 − 0.987i)8-s + (0.951 + 0.309i)10-s + (−0.896 − 1.76i)13-s + (−0.309 + 0.951i)16-s + (1.87 − 0.297i)17-s + (−0.707 − 0.707i)20-s + (0.951 − 0.309i)25-s + 1.97i·26-s + (1.44 − 1.04i)29-s + (0.707 − 0.707i)32-s + (−1.80 − 0.587i)34-s + (0.809 − 0.412i)37-s + ⋯

Functional equation

Λ(s)=(900s/2 ΓC(s) L(s)=((0.356+0.934i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(900s/2 ΓC(s) L(s)=((0.356+0.934i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 900900    =    22⋅32⋅522^{2} \cdot 3^{2} \cdot 5^{2}
Sign: 0.356+0.934i0.356 + 0.934i
Analytic conductor: 0.4491580.449158
Root analytic conductor: 0.6701920.670192
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ900(467,⋅)\chi_{900} (467, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 900, ( :0), 0.356+0.934i)(2,\ 900,\ (\ :0),\ 0.356 + 0.934i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.53703483590.5370348359
L(12)L(\frac12) ≈\approx 0.53703483590.5370348359
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+(0.891+0.453i)T 1 + (0.891 + 0.453i)T
3 1 1
5 1+(0.987−0.156i)T 1 + (0.987 - 0.156i)T
good7 1−iT2 1 - iT^{2}
11 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
13 1+(0.896+1.76i)T+(−0.587+0.809i)T2 1 + (0.896 + 1.76i)T + (-0.587 + 0.809i)T^{2}
17 1+(−1.87+0.297i)T+(0.951−0.309i)T2 1 + (-1.87 + 0.297i)T + (0.951 - 0.309i)T^{2}
19 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
23 1+(0.587+0.809i)T2 1 + (0.587 + 0.809i)T^{2}
29 1+(−1.44+1.04i)T+(0.309−0.951i)T2 1 + (-1.44 + 1.04i)T + (0.309 - 0.951i)T^{2}
31 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
37 1+(−0.809+0.412i)T+(0.587−0.809i)T2 1 + (-0.809 + 0.412i)T + (0.587 - 0.809i)T^{2}
41 1+(0.297+0.0966i)T+(0.809+0.587i)T2 1 + (0.297 + 0.0966i)T + (0.809 + 0.587i)T^{2}
43 1+iT2 1 + iT^{2}
47 1+(0.951+0.309i)T2 1 + (0.951 + 0.309i)T^{2}
53 1+(1.16+0.183i)T+(0.951+0.309i)T2 1 + (1.16 + 0.183i)T + (0.951 + 0.309i)T^{2}
59 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
61 1+(0.363+1.11i)T+(−0.809+0.587i)T2 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2}
67 1+(0.951−0.309i)T2 1 + (0.951 - 0.309i)T^{2}
71 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
73 1+(0.278+0.142i)T+(0.587+0.809i)T2 1 + (0.278 + 0.142i)T + (0.587 + 0.809i)T^{2}
79 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
83 1+(0.951−0.309i)T2 1 + (0.951 - 0.309i)T^{2}
89 1+(0.280+0.863i)T+(−0.809+0.587i)T2 1 + (0.280 + 0.863i)T + (-0.809 + 0.587i)T^{2}
97 1+(1.76+0.278i)T+(0.951+0.309i)T2 1 + (1.76 + 0.278i)T + (0.951 + 0.309i)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

−10.14187416871903901768840820021, −9.530535003300153030407508728559, −8.137920939482265566060771261649, −7.943759655829877788392517519113, −7.18340668232805453593223807273, −5.93040802236101095765148118835, −4.68538096177814200061650787726, −3.37061683995461990215091699819, −2.77181975301344430612239502277, −0.800737410943779763670657227828, 1.39622071123164531093632587568, 2.97849505773504656027527218061, 4.36930043618597602290040361502, 5.27320367621786531408599916330, 6.53103093691612159409917816161, 7.21039953847556823222320527991, 7.967875545285242817702062677209, 8.703030138584828463305306620173, 9.577687336697017679990293943432, 10.25533520458262273922590206509

Graph of the ZZ-function along the critical line