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Properties

Label 2-42e2-49.12-c0-0-0
Degree 22
Conductor 17641764
Sign 0.232−0.972i0.232 - 0.972i
Analytic cond. 0.8803500.880350
Root an. cond. 0.9382700.938270
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  + (−0.826 + 0.563i)7-s + (−0.590 + 1.22i)13-s + (0.975 + 0.563i)19-s + (0.826 + 0.563i)25-s + (−1.61 + 0.930i)31-s + (0.722 + 1.84i)37-s + (0.0332 − 0.145i)43-s + (0.365 − 0.930i)49-s + (1.45 − 0.571i)61-s + (−0.955 − 1.65i)67-s + (−0.167 + 0.246i)73-s + (−0.365 + 0.632i)79-s + (−0.202 − 1.34i)91-s + 0.867i·97-s + (−0.587 + 1.90i)103-s + ⋯
L(s)  = 1  + (−0.826 + 0.563i)7-s + (−0.590 + 1.22i)13-s + (0.975 + 0.563i)19-s + (0.826 + 0.563i)25-s + (−1.61 + 0.930i)31-s + (0.722 + 1.84i)37-s + (0.0332 − 0.145i)43-s + (0.365 − 0.930i)49-s + (1.45 − 0.571i)61-s + (−0.955 − 1.65i)67-s + (−0.167 + 0.246i)73-s + (−0.365 + 0.632i)79-s + (−0.202 − 1.34i)91-s + 0.867i·97-s + (−0.587 + 1.90i)103-s + ⋯

Functional equation

Λ(s)=(1764s/2 ΓC(s) L(s)=((0.232−0.972i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1764s/2 ΓC(s) L(s)=((0.232−0.972i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17641764    =    22⋅32⋅722^{2} \cdot 3^{2} \cdot 7^{2}
Sign: 0.232−0.972i0.232 - 0.972i
Analytic conductor: 0.8803500.880350
Root analytic conductor: 0.9382700.938270
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1764(649,⋅)\chi_{1764} (649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1764, ( :0), 0.232−0.972i)(2,\ 1764,\ (\ :0),\ 0.232 - 0.972i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.90326182480.9032618248
L(12)L(\frac12) ≈\approx 0.90326182480.9032618248
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 1
3 1 1
7 1+(0.826−0.563i)T 1 + (0.826 - 0.563i)T
good5 1+(−0.826−0.563i)T2 1 + (-0.826 - 0.563i)T^{2}
11 1+(−0.988+0.149i)T2 1 + (-0.988 + 0.149i)T^{2}
13 1+(0.590−1.22i)T+(−0.623−0.781i)T2 1 + (0.590 - 1.22i)T + (-0.623 - 0.781i)T^{2}
17 1+(−0.955−0.294i)T2 1 + (-0.955 - 0.294i)T^{2}
19 1+(−0.975−0.563i)T+(0.5+0.866i)T2 1 + (-0.975 - 0.563i)T + (0.5 + 0.866i)T^{2}
23 1+(0.955−0.294i)T2 1 + (0.955 - 0.294i)T^{2}
29 1+(−0.222−0.974i)T2 1 + (-0.222 - 0.974i)T^{2}
31 1+(1.61−0.930i)T+(0.5−0.866i)T2 1 + (1.61 - 0.930i)T + (0.5 - 0.866i)T^{2}
37 1+(−0.722−1.84i)T+(−0.733+0.680i)T2 1 + (-0.722 - 1.84i)T + (-0.733 + 0.680i)T^{2}
41 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
43 1+(−0.0332+0.145i)T+(−0.900−0.433i)T2 1 + (-0.0332 + 0.145i)T + (-0.900 - 0.433i)T^{2}
47 1+(−0.365+0.930i)T2 1 + (-0.365 + 0.930i)T^{2}
53 1+(−0.733−0.680i)T2 1 + (-0.733 - 0.680i)T^{2}
59 1+(−0.826+0.563i)T2 1 + (-0.826 + 0.563i)T^{2}
61 1+(−1.45+0.571i)T+(0.733−0.680i)T2 1 + (-1.45 + 0.571i)T + (0.733 - 0.680i)T^{2}
67 1+(0.955+1.65i)T+(−0.5+0.866i)T2 1 + (0.955 + 1.65i)T + (-0.5 + 0.866i)T^{2}
71 1+(−0.222+0.974i)T2 1 + (-0.222 + 0.974i)T^{2}
73 1+(0.167−0.246i)T+(−0.365−0.930i)T2 1 + (0.167 - 0.246i)T + (-0.365 - 0.930i)T^{2}
79 1+(0.365−0.632i)T+(−0.5−0.866i)T2 1 + (0.365 - 0.632i)T + (-0.5 - 0.866i)T^{2}
83 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
89 1+(0.988+0.149i)T2 1 + (0.988 + 0.149i)T^{2}
97 1−0.867iT−T2 1 - 0.867iT - 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

−9.476605675225854430690541183072, −9.109741304963965972019918059091, −8.126263108261862559686834295362, −7.10235969981292409880484890757, −6.60895600833690614797475677467, −5.58868580702850391724913819448, −4.84135292788876722116137899542, −3.67289543871433556881637552835, −2.84982457306577629642088138994, −1.63244375822373456471380590988, 0.69347759071025182469069164394, 2.48395941916319935296508840565, 3.32921518054907508555654711833, 4.27310679034682571741678712989, 5.37270272152770468975319676049, 6.01316272848023151044913984902, 7.29575311101666243852545916410, 7.38182767828418661917798789364, 8.602610128831082964533823011725, 9.437806640231741288607240951869

Graph of the ZZ-function along the critical line