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Properties

Label 2-3240-360.13-c0-0-5
Degree 22
Conductor 32403240
Sign −0.370+0.929i-0.370 + 0.929i
Analytic cond. 1.616971.61697
Root an. cond. 1.271601.27160
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  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.707 − 0.707i)5-s + (0.5 + 0.133i)7-s + (−0.707 + 0.707i)8-s + (−0.500 − 0.866i)10-s + (1.67 − 0.965i)11-s + (0.258 − 0.448i)14-s + (0.500 + 0.866i)16-s + (−0.965 + 0.258i)20-s + (−0.500 − 1.86i)22-s − 1.00i·25-s + (−0.366 − 0.366i)28-s + (0.707 + 1.22i)29-s + (−0.866 + 1.5i)31-s + (0.965 − 0.258i)32-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.707 − 0.707i)5-s + (0.5 + 0.133i)7-s + (−0.707 + 0.707i)8-s + (−0.500 − 0.866i)10-s + (1.67 − 0.965i)11-s + (0.258 − 0.448i)14-s + (0.500 + 0.866i)16-s + (−0.965 + 0.258i)20-s + (−0.500 − 1.86i)22-s − 1.00i·25-s + (−0.366 − 0.366i)28-s + (0.707 + 1.22i)29-s + (−0.866 + 1.5i)31-s + (0.965 − 0.258i)32-s + ⋯

Functional equation

Λ(s)=(3240s/2 ΓC(s) L(s)=((−0.370+0.929i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.370 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3240s/2 ΓC(s) L(s)=((−0.370+0.929i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.370 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32403240    =    23⋅34⋅52^{3} \cdot 3^{4} \cdot 5
Sign: −0.370+0.929i-0.370 + 0.929i
Analytic conductor: 1.616971.61697
Root analytic conductor: 1.271601.27160
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3240(2053,⋅)\chi_{3240} (2053, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3240, ( :0), −0.370+0.929i)(2,\ 3240,\ (\ :0),\ -0.370 + 0.929i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7003398451.700339845
L(12)L(\frac12) ≈\approx 1.7003398451.700339845
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+(−0.258+0.965i)T 1 + (-0.258 + 0.965i)T
3 1 1
5 1+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
good7 1+(−0.5−0.133i)T+(0.866+0.5i)T2 1 + (-0.5 - 0.133i)T + (0.866 + 0.5i)T^{2}
11 1+(−1.67+0.965i)T+(0.5−0.866i)T2 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2}
13 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
17 1−iT2 1 - iT^{2}
19 1+T2 1 + T^{2}
23 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
29 1+(−0.707−1.22i)T+(−0.5+0.866i)T2 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2}
31 1+(0.866−1.5i)T+(−0.5−0.866i)T2 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2}
37 1+iT2 1 + iT^{2}
41 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
43 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
47 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
53 1+(−1.22+1.22i)T−iT2 1 + (-1.22 + 1.22i)T - iT^{2}
59 1+(0.707−1.22i)T+(−0.5−0.866i)T2 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2}
61 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
67 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
71 1+T2 1 + T^{2}
73 1+(1.36+1.36i)T+iT2 1 + (1.36 + 1.36i)T + iT^{2}
79 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
83 1+(0.965+0.258i)T+(0.866+0.5i)T2 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2}
89 1−T2 1 - T^{2}
97 1+(1.86+0.5i)T+(0.866+0.5i)T2 1 + (1.86 + 0.5i)T + (0.866 + 0.5i)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

−8.813731767521998069212128514669, −8.374875841993555022261963025776, −6.93553957368096726080837164592, −6.11140706427783096517832429796, −5.39043950170534651249225898683, −4.70828893392109529575319335751, −3.83001846640959255791206407784, −3.00488362848851984340152916119, −1.71129099332832130864366308904, −1.16460416644912233229130136477, 1.50800330657816154851757514048, 2.67604987064960688685270688112, 3.95510401535467828059395678673, 4.36647179121525449452429474301, 5.47308205827717030307496672650, 6.17275011679719354234983655155, 6.78125502316049827651305751848, 7.38468674399473646198366808685, 8.137346885147345462425270856844, 9.147057973088083393062506384053

Graph of the ZZ-function along the critical line