↑

Properties

Label 2-1200-15.14-c0-0-0
Degree 22
Conductor 12001200
Sign −0.447−0.894i-0.447 - 0.894i
Analytic cond. 0.5988780.598878
Root an. cond. 0.7738720.773872
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  + i·3-s + i·7-s − 9-s + i·13-s − 19-s − 21-s − i·27-s + 31-s + 2i·37-s − 39-s − i·43-s − i·57-s − 61-s − i·63-s + i·67-s + ⋯
L(s)  = 1  + i·3-s + i·7-s − 9-s + i·13-s − 19-s − 21-s − i·27-s + 31-s + 2i·37-s − 39-s − i·43-s − i·57-s − 61-s − i·63-s + i·67-s + ⋯

Functional equation

Λ(s)=(1200s/2 ΓC(s) L(s)=((−0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1200s/2 ΓC(s) L(s)=((−0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12001200    =    24⋅3⋅522^{4} \cdot 3 \cdot 5^{2}
Sign: −0.447−0.894i-0.447 - 0.894i
Analytic conductor: 0.5988780.598878
Root analytic conductor: 0.7738720.773872
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1200(449,⋅)\chi_{1200} (449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1200, ( :0), −0.447−0.894i)(2,\ 1200,\ (\ :0),\ -0.447 - 0.894i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.92365890070.9236589007
L(12)L(\frac12) ≈\approx 0.92365890070.9236589007
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
3 1−iT 1 - iT
5 1 1
good7 1−iT−T2 1 - iT - T^{2}
11 1−T2 1 - T^{2}
13 1−iT−T2 1 - iT - T^{2}
17 1+T2 1 + T^{2}
19 1+T+T2 1 + T + T^{2}
23 1+T2 1 + T^{2}
29 1−T2 1 - T^{2}
31 1−T+T2 1 - T + T^{2}
37 1−2iT−T2 1 - 2iT - T^{2}
41 1−T2 1 - T^{2}
43 1+iT−T2 1 + iT - T^{2}
47 1+T2 1 + T^{2}
53 1+T2 1 + T^{2}
59 1−T2 1 - T^{2}
61 1+T+T2 1 + T + T^{2}
67 1−iT−T2 1 - iT - T^{2}
71 1−T2 1 - T^{2}
73 1+2iT−T2 1 + 2iT - T^{2}
79 1−2T+T2 1 - 2T + T^{2}
83 1+T2 1 + T^{2}
89 1−T2 1 - T^{2}
97 1+iT−T2 1 + iT - 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

−10.16692126896792320217436702249, −9.324223472328552274411279417898, −8.760434694483442977948695514813, −8.076622587861965813696973815329, −6.66021730851117044885065126857, −5.99161771659389753301739223094, −4.96767717499576217254459612318, −4.28823171968519359752415878042, −3.14018841927764738210195200695, −2.10863678667909022249764221974, 0.817156158108819163320116772205, 2.22942052793984426763547218566, 3.36691859714621788828279753969, 4.48280741826464614963215358653, 5.65924444420486887252769112857, 6.44497921649453797621235043873, 7.28822144623843650417256240064, 7.915532844020275280223287750422, 8.637800860934582289020213754342, 9.712988721814415946608443269753

Graph of the ZZ-function along the critical line