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Properties

Label 2-3960-440.109-c0-0-4
Degree 22
Conductor 39603960
Sign 0.707−0.707i0.707 - 0.707i
Analytic cond. 1.976291.97629
Root an. cond. 1.405801.40580
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  + (−0.707 − 0.707i)2-s + 1.00i·4-s + i·5-s + 1.41·7-s + (0.707 − 0.707i)8-s + (0.707 − 0.707i)10-s + i·11-s + 1.41i·13-s + (−1.00 − 1.00i)14-s − 1.00·16-s + 1.41·17-s − 1.00·20-s + (0.707 − 0.707i)22-s − 25-s + (1.00 − 1.00i)26-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + i·5-s + 1.41·7-s + (0.707 − 0.707i)8-s + (0.707 − 0.707i)10-s + i·11-s + 1.41i·13-s + (−1.00 − 1.00i)14-s − 1.00·16-s + 1.41·17-s − 1.00·20-s + (0.707 − 0.707i)22-s − 25-s + (1.00 − 1.00i)26-s + ⋯

Functional equation

Λ(s)=(3960s/2 ΓC(s) L(s)=((0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3960s/2 ΓC(s) L(s)=((0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39603960    =    23⋅32⋅5⋅112^{3} \cdot 3^{2} \cdot 5 \cdot 11
Sign: 0.707−0.707i0.707 - 0.707i
Analytic conductor: 1.976291.97629
Root analytic conductor: 1.405801.40580
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3960(109,⋅)\chi_{3960} (109, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3960, ( :0), 0.707−0.707i)(2,\ 3960,\ (\ :0),\ 0.707 - 0.707i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0928954941.092895494
L(12)L(\frac12) ≈\approx 1.0928954941.092895494
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+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
3 1 1
5 1−iT 1 - iT
11 1−iT 1 - iT
good7 1−1.41T+T2 1 - 1.41T + T^{2}
13 1−1.41iT−T2 1 - 1.41iT - T^{2}
17 1−1.41T+T2 1 - 1.41T + T^{2}
19 1+T2 1 + T^{2}
23 1−T2 1 - T^{2}
29 1+T2 1 + T^{2}
31 1+T2 1 + T^{2}
37 1+T2 1 + T^{2}
41 1−T2 1 - T^{2}
43 1+1.41iT−T2 1 + 1.41iT - T^{2}
47 1−T2 1 - T^{2}
53 1+T2 1 + T^{2}
59 1−T2 1 - T^{2}
61 1+T2 1 + T^{2}
67 1+T2 1 + T^{2}
71 1+2T+T2 1 + 2T + T^{2}
73 1+1.41T+T2 1 + 1.41T + T^{2}
79 1−T2 1 - T^{2}
83 1+1.41iT−T2 1 + 1.41iT - T^{2}
89 1+T2 1 + T^{2}
97 1−T2 1 - 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

−8.780587617986931920600691956584, −7.988816694162300191061301765304, −7.32160880544288757699742539018, −6.99093144867118979164578867771, −5.79746747482598588582334750793, −4.67703488420814108205298399017, −4.08883051632040673056338765191, −3.10205122068549285682381847156, −2.04642552366789615559020104049, −1.56087167796996927417834091876, 0.878954898968080740854605256906, 1.52864835997380095223083001340, 2.97022727409248490972132171897, 4.25804450884861233973253832055, 5.16052783695257121279945934276, 5.50746978455129918063050764395, 6.17021967577763715367775231448, 7.58032594095621972620311549858, 7.85946271342171567073760242043, 8.409565045463735843561576414732

Graph of the ZZ-function along the critical line