↑

Properties

Label 2-30e2-60.47-c0-0-0
Degree 22
Conductor 900900
Sign 0.920−0.391i0.920 - 0.391i
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)8-s + (1 − i)13-s − 1.00·16-s + 1.41i·26-s + 1.41·29-s + (0.707 − 0.707i)32-s + (1 + i)37-s − 1.41i·41-s + i·49-s + (−1.00 − 1.00i)52-s + (−1.00 + 1.00i)58-s + 1.00i·64-s + (−1 + i)73-s − 1.41·74-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)8-s + (1 − i)13-s − 1.00·16-s + 1.41i·26-s + 1.41·29-s + (0.707 − 0.707i)32-s + (1 + i)37-s − 1.41i·41-s + i·49-s + (−1.00 − 1.00i)52-s + (−1.00 + 1.00i)58-s + 1.00i·64-s + (−1 + i)73-s − 1.41·74-s + ⋯

Functional equation

Λ(s)=(900s/2 ΓC(s) L(s)=((0.920−0.391i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(900s/2 ΓC(s) L(s)=((0.920−0.391i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 900900    =    22⋅32⋅522^{2} \cdot 3^{2} \cdot 5^{2}
Sign: 0.920−0.391i0.920 - 0.391i
Analytic conductor: 0.4491580.449158
Root analytic conductor: 0.6701920.670192
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ900(107,⋅)\chi_{900} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 900, ( :0), 0.920−0.391i)(2,\ 900,\ (\ :0),\ 0.920 - 0.391i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.73802698370.7380269837
L(12)L(\frac12) ≈\approx 0.73802698370.7380269837
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+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
3 1 1
5 1 1
good7 1−iT2 1 - iT^{2}
11 1+T2 1 + T^{2}
13 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
17 1−iT2 1 - iT^{2}
19 1+T2 1 + T^{2}
23 1−iT2 1 - iT^{2}
29 1−1.41T+T2 1 - 1.41T + T^{2}
31 1−T2 1 - T^{2}
37 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
41 1+1.41iT−T2 1 + 1.41iT - T^{2}
43 1+iT2 1 + iT^{2}
47 1+iT2 1 + iT^{2}
53 1+iT2 1 + iT^{2}
59 1−T2 1 - T^{2}
61 1+T2 1 + T^{2}
67 1−iT2 1 - iT^{2}
71 1+T2 1 + T^{2}
73 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
79 1+T2 1 + T^{2}
83 1−iT2 1 - iT^{2}
89 1+1.41T+T2 1 + 1.41T + T^{2}
97 1+(1+i)T+iT2 1 + (1 + i)T + iT^{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

−10.31306200753245864371643572220, −9.438723652618452412335122686870, −8.503013309407695348192527299584, −8.023245722412576788004270925851, −7.00046957759112566052959720995, −6.14777995842846822443576016168, −5.40395388392271782895369443928, −4.27499228808693724421674533876, −2.81387087182950764640514526611, −1.17460419426996029119861987527, 1.35560445489355166254832841367, 2.62922627588755500792718818633, 3.78105983815928603619611147719, 4.64550398783395833253484122584, 6.15759558850442512729912174432, 6.96567193406411933816533276834, 8.008155191505087720301018702208, 8.703023568523237304613196399376, 9.445698443152276969134315994917, 10.24303806571410968256964277428

Graph of the ZZ-function along the critical line