↑

Properties

Label 2-3380-3380.223-c0-0-0
Degree 22
Conductor 33803380
Sign 0.998−0.0598i0.998 - 0.0598i
Analytic cond. 1.686831.68683
Root an. cond. 1.298781.29878
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  + (0.903 + 0.428i)2-s + (0.632 + 0.774i)4-s + (0.692 − 0.721i)5-s + (0.239 + 0.970i)8-s + (−0.0804 − 0.996i)9-s + (0.935 − 0.354i)10-s + (−0.278 − 0.960i)13-s + (−0.200 + 0.979i)16-s + (−0.176 + 0.0969i)17-s + (0.354 − 0.935i)18-s + (0.996 + 0.0804i)20-s + (−0.0402 − 0.999i)25-s + (0.160 − 0.987i)26-s + (1.52 + 0.724i)29-s + (−0.600 + 0.799i)32-s + ⋯
L(s)  = 1  + (0.903 + 0.428i)2-s + (0.632 + 0.774i)4-s + (0.692 − 0.721i)5-s + (0.239 + 0.970i)8-s + (−0.0804 − 0.996i)9-s + (0.935 − 0.354i)10-s + (−0.278 − 0.960i)13-s + (−0.200 + 0.979i)16-s + (−0.176 + 0.0969i)17-s + (0.354 − 0.935i)18-s + (0.996 + 0.0804i)20-s + (−0.0402 − 0.999i)25-s + (0.160 − 0.987i)26-s + (1.52 + 0.724i)29-s + (−0.600 + 0.799i)32-s + ⋯

Functional equation

Λ(s)=(3380s/2 ΓC(s) L(s)=((0.998−0.0598i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0598i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3380s/2 ΓC(s) L(s)=((0.998−0.0598i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0598i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33803380    =    22⋅5⋅1322^{2} \cdot 5 \cdot 13^{2}
Sign: 0.998−0.0598i0.998 - 0.0598i
Analytic conductor: 1.686831.68683
Root analytic conductor: 1.298781.29878
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3380(223,⋅)\chi_{3380} (223, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3380, ( :0), 0.998−0.0598i)(2,\ 3380,\ (\ :0),\ 0.998 - 0.0598i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.4686881572.468688157
L(12)L(\frac12) ≈\approx 2.4686881572.468688157
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+(−0.903−0.428i)T 1 + (-0.903 - 0.428i)T
5 1+(−0.692+0.721i)T 1 + (-0.692 + 0.721i)T
13 1+(0.278+0.960i)T 1 + (0.278 + 0.960i)T
good3 1+(0.0804+0.996i)T2 1 + (0.0804 + 0.996i)T^{2}
7 1+(0.845+0.534i)T2 1 + (0.845 + 0.534i)T^{2}
11 1+(0.160−0.987i)T2 1 + (0.160 - 0.987i)T^{2}
17 1+(0.176−0.0969i)T+(0.534−0.845i)T2 1 + (0.176 - 0.0969i)T + (0.534 - 0.845i)T^{2}
19 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
23 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
29 1+(−1.52−0.724i)T+(0.632+0.774i)T2 1 + (-1.52 - 0.724i)T + (0.632 + 0.774i)T^{2}
31 1+(−0.239−0.970i)T2 1 + (-0.239 - 0.970i)T^{2}
37 1+(−0.506+0.380i)T+(0.278−0.960i)T2 1 + (-0.506 + 0.380i)T + (0.278 - 0.960i)T^{2}
41 1+(−1.38−1.27i)T+(0.0804+0.996i)T2 1 + (-1.38 - 1.27i)T + (0.0804 + 0.996i)T^{2}
43 1+(−0.960+0.278i)T2 1 + (-0.960 + 0.278i)T^{2}
47 1+(0.748−0.663i)T2 1 + (0.748 - 0.663i)T^{2}
53 1+(0.825+0.499i)T+(0.464+0.885i)T2 1 + (0.825 + 0.499i)T + (0.464 + 0.885i)T^{2}
59 1+(−0.391−0.919i)T2 1 + (-0.391 - 0.919i)T^{2}
61 1+(1.38−1.44i)T+(−0.0402−0.999i)T2 1 + (1.38 - 1.44i)T + (-0.0402 - 0.999i)T^{2}
67 1+(−0.200−0.979i)T2 1 + (-0.200 - 0.979i)T^{2}
71 1+(0.903+0.428i)T2 1 + (0.903 + 0.428i)T^{2}
73 1+(0.822+0.568i)T+(0.354+0.935i)T2 1 + (0.822 + 0.568i)T + (0.354 + 0.935i)T^{2}
79 1+(−0.748+0.663i)T2 1 + (-0.748 + 0.663i)T^{2}
83 1+(−0.568+0.822i)T2 1 + (-0.568 + 0.822i)T^{2}
89 1+(1.92+0.516i)T+(0.866+0.5i)T2 1 + (1.92 + 0.516i)T + (0.866 + 0.5i)T^{2}
97 1+(−0.628−1.88i)T+(−0.799+0.600i)T2 1 + (-0.628 - 1.88i)T + (-0.799 + 0.600i)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

−8.654951067993459604627185380839, −8.046857794417080164976848795107, −7.15670474886417504101142902522, −6.23707450903177296309758478383, −5.91651958780361044021295072179, −4.94918250757423534838949596199, −4.40384092068913056899126077970, −3.28936771675559231388682190185, −2.58165295873418800491195823970, −1.22175820384110944478128283519, 1.61340399972871653523371422267, 2.41493731781200731506890920457, 3.04672685704762178504521973879, 4.29877262869242789485202923034, 4.81395154256590254696239841119, 5.79709577197059398577045715831, 6.35233166212696057334351536172, 7.10597335771036114395258723514, 7.81333289126784137970893210905, 8.976522683302745381414792757321

Graph of the ZZ-function along the critical line