↑

Properties

Label 2-3040-3040.2469-c0-0-2
Degree 22
Conductor 30403040
Sign 0.634+0.773i0.634 + 0.773i
Analytic cond. 1.517151.51715
Root an. cond. 1.231721.23172
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  + (−0.634 + 0.773i)2-s + (−0.761 − 1.83i)3-s + (−0.195 − 0.980i)4-s + (−0.923 − 0.382i)5-s + (1.90 + 0.577i)6-s + (0.881 + 0.471i)8-s + (−2.09 + 2.09i)9-s + (0.881 − 0.471i)10-s + (−0.425 + 1.02i)11-s + (−1.65 + 1.10i)12-s + (1.76 − 0.732i)13-s + 1.99i·15-s + (−0.923 + 0.382i)16-s + (−0.290 − 2.94i)18-s + (0.923 − 0.382i)19-s + (−0.195 + 0.980i)20-s + ⋯
L(s)  = 1  + (−0.634 + 0.773i)2-s + (−0.761 − 1.83i)3-s + (−0.195 − 0.980i)4-s + (−0.923 − 0.382i)5-s + (1.90 + 0.577i)6-s + (0.881 + 0.471i)8-s + (−2.09 + 2.09i)9-s + (0.881 − 0.471i)10-s + (−0.425 + 1.02i)11-s + (−1.65 + 1.10i)12-s + (1.76 − 0.732i)13-s + 1.99i·15-s + (−0.923 + 0.382i)16-s + (−0.290 − 2.94i)18-s + (0.923 − 0.382i)19-s + (−0.195 + 0.980i)20-s + ⋯

Functional equation

Λ(s)=(3040s/2 ΓC(s) L(s)=((0.634+0.773i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3040s/2 ΓC(s) L(s)=((0.634+0.773i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 30403040    =    25⋅5⋅192^{5} \cdot 5 \cdot 19
Sign: 0.634+0.773i0.634 + 0.773i
Analytic conductor: 1.517151.51715
Root analytic conductor: 1.231721.23172
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3040(2469,⋅)\chi_{3040} (2469, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3040, ( :0), 0.634+0.773i)(2,\ 3040,\ (\ :0),\ 0.634 + 0.773i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.54475892130.5447589213
L(12)L(\frac12) ≈\approx 0.54475892130.5447589213
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+(0.634−0.773i)T 1 + (0.634 - 0.773i)T
5 1+(0.923+0.382i)T 1 + (0.923 + 0.382i)T
19 1+(−0.923+0.382i)T 1 + (-0.923 + 0.382i)T
good3 1+(0.761+1.83i)T+(−0.707+0.707i)T2 1 + (0.761 + 1.83i)T + (-0.707 + 0.707i)T^{2}
7 1−iT2 1 - iT^{2}
11 1+(0.425−1.02i)T+(−0.707−0.707i)T2 1 + (0.425 - 1.02i)T + (-0.707 - 0.707i)T^{2}
13 1+(−1.76+0.732i)T+(0.707−0.707i)T2 1 + (-1.76 + 0.732i)T + (0.707 - 0.707i)T^{2}
17 1+T2 1 + T^{2}
23 1+iT2 1 + iT^{2}
29 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
31 1−T2 1 - T^{2}
37 1+(−1.42−0.591i)T+(0.707+0.707i)T2 1 + (-1.42 - 0.591i)T + (0.707 + 0.707i)T^{2}
41 1+iT2 1 + iT^{2}
43 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
47 1+T2 1 + T^{2}
53 1+(0.0750−0.181i)T+(−0.707−0.707i)T2 1 + (0.0750 - 0.181i)T + (-0.707 - 0.707i)T^{2}
59 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
61 1+(−0.636−1.53i)T+(−0.707+0.707i)T2 1 + (-0.636 - 1.53i)T + (-0.707 + 0.707i)T^{2}
67 1+(0.360+0.871i)T+(−0.707+0.707i)T2 1 + (0.360 + 0.871i)T + (-0.707 + 0.707i)T^{2}
71 1−iT2 1 - iT^{2}
73 1+iT2 1 + iT^{2}
79 1+T2 1 + T^{2}
83 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
89 1−iT2 1 - iT^{2}
97 1+0.580T+T2 1 + 0.580T + 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

−8.348089022644211656480812814207, −7.87584265132694273093091371556, −7.40218567605663062476126298689, −6.72424995686711870912459551053, −5.95381281395542512151601277358, −5.36217698685093026391277251531, −4.47464756921752836221533359978, −2.85692441778528103186226559464, −1.50237810312977994272914693728, −0.807037847904901476122377726525, 0.78328784429321338777483151989, 2.89357736058999935334140994805, 3.69479320398835847888327842171, 3.88275861732806277778839331272, 4.89611203997275863954376491272, 5.86731992245014642298779141204, 6.62057110949919508909405200270, 7.919393956986898076986998417837, 8.569841839124860644529063425440, 9.095263969293070132263730391044

Graph of the ZZ-function along the critical line