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Properties

Label 2-3380-3380.1783-c0-0-0
Degree 22
Conductor 33803380
Sign 0.405−0.914i0.405 - 0.914i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.534 − 0.845i)2-s + (−0.428 − 0.903i)4-s + (−0.919 − 0.391i)5-s + (−0.992 − 0.120i)8-s + (−0.999 − 0.0402i)9-s + (−0.822 + 0.568i)10-s + (−0.799 + 0.600i)13-s + (−0.632 + 0.774i)16-s + (0.0943 + 0.664i)17-s + (−0.568 + 0.822i)18-s + (0.0402 + 0.999i)20-s + (0.692 + 0.721i)25-s + (0.0804 + 0.996i)26-s + (−0.297 + 0.470i)29-s + (0.316 + 0.948i)32-s + ⋯
L(s)  = 1  + (0.534 − 0.845i)2-s + (−0.428 − 0.903i)4-s + (−0.919 − 0.391i)5-s + (−0.992 − 0.120i)8-s + (−0.999 − 0.0402i)9-s + (−0.822 + 0.568i)10-s + (−0.799 + 0.600i)13-s + (−0.632 + 0.774i)16-s + (0.0943 + 0.664i)17-s + (−0.568 + 0.822i)18-s + (0.0402 + 0.999i)20-s + (0.692 + 0.721i)25-s + (0.0804 + 0.996i)26-s + (−0.297 + 0.470i)29-s + (0.316 + 0.948i)32-s + ⋯

Functional equation

Λ(s)=(3380s/2 ΓC(s) L(s)=((0.405−0.914i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.405 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3380s/2 ΓC(s) L(s)=((0.405−0.914i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.405 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33803380    =    22⋅5⋅1322^{2} \cdot 5 \cdot 13^{2}
Sign: 0.405−0.914i0.405 - 0.914i
Analytic conductor: 1.686831.68683
Root analytic conductor: 1.298781.29878
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3380(1783,⋅)\chi_{3380} (1783, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3380, ( :0), 0.405−0.914i)(2,\ 3380,\ (\ :0),\ 0.405 - 0.914i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.27576852880.2757685288
L(12)L(\frac12) ≈\approx 0.27576852880.2757685288
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+(−0.534+0.845i)T 1 + (-0.534 + 0.845i)T
5 1+(0.919+0.391i)T 1 + (0.919 + 0.391i)T
13 1+(0.799−0.600i)T 1 + (0.799 - 0.600i)T
good3 1+(0.999+0.0402i)T2 1 + (0.999 + 0.0402i)T^{2}
7 1+(−0.278−0.960i)T2 1 + (-0.278 - 0.960i)T^{2}
11 1+(0.0804+0.996i)T2 1 + (0.0804 + 0.996i)T^{2}
17 1+(−0.0943−0.664i)T+(−0.960+0.278i)T2 1 + (-0.0943 - 0.664i)T + (-0.960 + 0.278i)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+(0.297−0.470i)T+(−0.428−0.903i)T2 1 + (0.297 - 0.470i)T + (-0.428 - 0.903i)T^{2}
31 1+(0.992+0.120i)T2 1 + (0.992 + 0.120i)T^{2}
37 1+(0.304+0.101i)T+(0.799+0.600i)T2 1 + (0.304 + 0.101i)T + (0.799 + 0.600i)T^{2}
41 1+(−0.439−0.00884i)T+(0.999+0.0402i)T2 1 + (-0.439 - 0.00884i)T + (0.999 + 0.0402i)T^{2}
43 1+(0.600+0.799i)T2 1 + (0.600 + 0.799i)T^{2}
47 1+(0.354−0.935i)T2 1 + (0.354 - 0.935i)T^{2}
53 1+(1.55−1.22i)T+(0.239−0.970i)T2 1 + (1.55 - 1.22i)T + (0.239 - 0.970i)T^{2}
59 1+(−0.979−0.200i)T2 1 + (-0.979 - 0.200i)T^{2}
61 1+(1.32+0.565i)T+(0.692+0.721i)T2 1 + (1.32 + 0.565i)T + (0.692 + 0.721i)T^{2}
67 1+(−0.632−0.774i)T2 1 + (-0.632 - 0.774i)T^{2}
71 1+(0.534−0.845i)T2 1 + (0.534 - 0.845i)T^{2}
73 1+(−0.464+0.885i)T+(−0.568−0.822i)T2 1 + (-0.464 + 0.885i)T + (-0.568 - 0.822i)T^{2}
79 1+(−0.354+0.935i)T2 1 + (-0.354 + 0.935i)T^{2}
83 1+(−0.885−0.464i)T2 1 + (-0.885 - 0.464i)T^{2}
89 1+(1.76+0.472i)T+(0.866+0.5i)T2 1 + (1.76 + 0.472i)T + (0.866 + 0.5i)T^{2}
97 1+(0.212−1.30i)T+(−0.948−0.316i)T2 1 + (0.212 - 1.30i)T + (-0.948 - 0.316i)T^{2}
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   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

−9.114351585691209625562781983198, −8.299078555589038374759757508558, −7.55524780836600664312530674609, −6.52591888500822064057927379423, −5.71752768995417369536250245546, −4.90495805437735642310361038498, −4.26092824489324639468597860051, −3.40897122655715972357018525892, −2.63647607182236538811008897938, −1.44588185845495715952274669371, 0.13446769328246180923009382043, 2.59540721424033198790809179145, 3.17812570525755094307484765428, 4.06920342430244478917576531504, 4.96703567747096738116153353402, 5.56598451563630862858377103811, 6.48855944822224781540284585039, 7.18154108047006003339616111638, 7.86227291349928958228087574631, 8.317601498047720360939864515232

Graph of the ZZ-function along the critical line