↑

Properties

Label 2-126-7.4-c5-0-5
Degree 22
Conductor 126126
Sign −0.283−0.958i-0.283 - 0.958i
Analytic cond. 20.208320.2083
Root an. cond. 4.495374.49537
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 3.46i)2-s + (−7.99 + 13.8i)4-s + (−37.7 − 65.3i)5-s + (99.4 + 83.1i)7-s − 63.9·8-s + (150. − 261. i)10-s + (−74.7 + 129. i)11-s + 349.·13-s + (−88.9 + 510. i)14-s + (−128 − 221. i)16-s + (−574. + 995. i)17-s + (1.39e3 + 2.42e3i)19-s + 1.20e3·20-s − 597.·22-s + (906. + 1.57e3i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.675 − 1.16i)5-s + (0.767 + 0.641i)7-s − 0.353·8-s + (0.477 − 0.826i)10-s + (−0.186 + 0.322i)11-s + 0.573·13-s + (−0.121 + 0.696i)14-s + (−0.125 − 0.216i)16-s + (−0.482 + 0.835i)17-s + (0.888 + 1.53i)19-s + 0.675·20-s − 0.263·22-s + (0.357 + 0.619i)23-s + ⋯

Functional equation

Λ(s)=(126s/2 ΓC(s) L(s)=((−0.283−0.958i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.283 - 0.958i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(126s/2 ΓC(s+5/2) L(s)=((−0.283−0.958i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.283 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 126126    =    2⋅32⋅72 \cdot 3^{2} \cdot 7
Sign: −0.283−0.958i-0.283 - 0.958i
Analytic conductor: 20.208320.2083
Root analytic conductor: 4.495374.49537
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ126(109,⋅)\chi_{126} (109, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 126, ( :5/2), −0.283−0.958i)(2,\ 126,\ (\ :5/2),\ -0.283 - 0.958i)

Particular Values

L(3)L(3) ≈\approx 1.7744374891.774437489
L(12)L(\frac12) ≈\approx 1.7744374891.774437489
L(72)L(\frac{7}{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+(−2−3.46i)T 1 + (-2 - 3.46i)T
3 1 1
7 1+(−99.4−83.1i)T 1 + (-99.4 - 83.1i)T
good5 1+(37.7+65.3i)T+(−1.56e3+2.70e3i)T2 1 + (37.7 + 65.3i)T + (-1.56e3 + 2.70e3i)T^{2}
11 1+(74.7−129.i)T+(−8.05e4−1.39e5i)T2 1 + (74.7 - 129. i)T + (-8.05e4 - 1.39e5i)T^{2}
13 1−349.T+3.71e5T2 1 - 349.T + 3.71e5T^{2}
17 1+(574.−995.i)T+(−7.09e5−1.22e6i)T2 1 + (574. - 995. i)T + (-7.09e5 - 1.22e6i)T^{2}
19 1+(−1.39e3−2.42e3i)T+(−1.23e6+2.14e6i)T2 1 + (-1.39e3 - 2.42e3i)T + (-1.23e6 + 2.14e6i)T^{2}
23 1+(−906.−1.57e3i)T+(−3.21e6+5.57e6i)T2 1 + (-906. - 1.57e3i)T + (-3.21e6 + 5.57e6i)T^{2}
29 1−759.T+2.05e7T2 1 - 759.T + 2.05e7T^{2}
31 1+(4.51e3−7.82e3i)T+(−1.43e7−2.47e7i)T2 1 + (4.51e3 - 7.82e3i)T + (-1.43e7 - 2.47e7i)T^{2}
37 1+(3.89e3+6.75e3i)T+(−3.46e7+6.00e7i)T2 1 + (3.89e3 + 6.75e3i)T + (-3.46e7 + 6.00e7i)T^{2}
41 1+7.64e3T+1.15e8T2 1 + 7.64e3T + 1.15e8T^{2}
43 1−1.21e4T+1.47e8T2 1 - 1.21e4T + 1.47e8T^{2}
47 1+(−1.22e4−2.13e4i)T+(−1.14e8+1.98e8i)T2 1 + (-1.22e4 - 2.13e4i)T + (-1.14e8 + 1.98e8i)T^{2}
53 1+(−6.79e3+1.17e4i)T+(−2.09e8−3.62e8i)T2 1 + (-6.79e3 + 1.17e4i)T + (-2.09e8 - 3.62e8i)T^{2}
59 1+(1.31e4−2.28e4i)T+(−3.57e8−6.19e8i)T2 1 + (1.31e4 - 2.28e4i)T + (-3.57e8 - 6.19e8i)T^{2}
61 1+(1.76e4+3.05e4i)T+(−4.22e8+7.31e8i)T2 1 + (1.76e4 + 3.05e4i)T + (-4.22e8 + 7.31e8i)T^{2}
67 1+(2.71e4−4.70e4i)T+(−6.75e8−1.16e9i)T2 1 + (2.71e4 - 4.70e4i)T + (-6.75e8 - 1.16e9i)T^{2}
71 1−7.01e4T+1.80e9T2 1 - 7.01e4T + 1.80e9T^{2}
73 1+(−2.22e4+3.85e4i)T+(−1.03e9−1.79e9i)T2 1 + (-2.22e4 + 3.85e4i)T + (-1.03e9 - 1.79e9i)T^{2}
79 1+(3.08e4+5.33e4i)T+(−1.53e9+2.66e9i)T2 1 + (3.08e4 + 5.33e4i)T + (-1.53e9 + 2.66e9i)T^{2}
83 1−8.71e4T+3.93e9T2 1 - 8.71e4T + 3.93e9T^{2}
89 1+(−4.92e4−8.53e4i)T+(−2.79e9+4.83e9i)T2 1 + (-4.92e4 - 8.53e4i)T + (-2.79e9 + 4.83e9i)T^{2}
97 1−3.23e4T+8.58e9T2 1 - 3.23e4T + 8.58e9T^{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

−12.53446872502082297497047132326, −12.16731854174008987027063596343, −10.86875413472673608775950323597, −9.146089291219129019229596722986, −8.370005846194992190988347012092, −7.54417407631467050454184774319, −5.81834115349266830926858240154, −4.91503309142647855348890016267, −3.74275675750012345361824928925, −1.46635481624463237676080239482, 0.61348372369356748237067130245, 2.58068510479999918438822279089, 3.76025057644362189423688096054, 5.02111176747107316413682331182, 6.73206824287478074101875780849, 7.65066742977170227177663064754, 9.074350102716754245921414724597, 10.54997256087262906756088590123, 11.15636219001585365005270904404, 11.75177074711858353578277098089

Graph of the ZZ-function along the critical line