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Properties

Label 2-42e2-49.3-c0-0-0
Degree 22
Conductor 17641764
Sign −0.274−0.961i-0.274 - 0.961i
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  + (−0.365 + 0.930i)7-s + (−1.55 + 1.24i)13-s + (−1.61 + 0.930i)19-s + (0.365 + 0.930i)25-s + (1.17 + 0.680i)31-s + (1.40 − 1.29i)37-s + (−1.78 − 0.858i)43-s + (−0.733 − 0.680i)49-s + (1.32 + 1.42i)61-s + (−0.826 + 1.43i)67-s + (0.548 − 0.215i)73-s + (0.733 + 1.26i)79-s + (−0.587 − 1.90i)91-s − 1.56i·97-s + (−0.167 + 0.246i)103-s + ⋯
L(s)  = 1  + (−0.365 + 0.930i)7-s + (−1.55 + 1.24i)13-s + (−1.61 + 0.930i)19-s + (0.365 + 0.930i)25-s + (1.17 + 0.680i)31-s + (1.40 − 1.29i)37-s + (−1.78 − 0.858i)43-s + (−0.733 − 0.680i)49-s + (1.32 + 1.42i)61-s + (−0.826 + 1.43i)67-s + (0.548 − 0.215i)73-s + (0.733 + 1.26i)79-s + (−0.587 − 1.90i)91-s − 1.56i·97-s + (−0.167 + 0.246i)103-s + ⋯

Functional equation

Λ(s)=(1764s/2 ΓC(s) L(s)=((−0.274−0.961i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1764s/2 ΓC(s) L(s)=((−0.274−0.961i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17641764    =    22⋅32⋅722^{2} \cdot 3^{2} \cdot 7^{2}
Sign: −0.274−0.961i-0.274 - 0.961i
Analytic conductor: 0.8803500.880350
Root analytic conductor: 0.9382700.938270
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1764(1081,⋅)\chi_{1764} (1081, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1764, ( :0), −0.274−0.961i)(2,\ 1764,\ (\ :0),\ -0.274 - 0.961i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.76532370190.7653237019
L(12)L(\frac12) ≈\approx 0.76532370190.7653237019
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+(0.365−0.930i)T 1 + (0.365 - 0.930i)T
good5 1+(−0.365−0.930i)T2 1 + (-0.365 - 0.930i)T^{2}
11 1+(0.955−0.294i)T2 1 + (0.955 - 0.294i)T^{2}
13 1+(1.55−1.24i)T+(0.222−0.974i)T2 1 + (1.55 - 1.24i)T + (0.222 - 0.974i)T^{2}
17 1+(−0.826−0.563i)T2 1 + (-0.826 - 0.563i)T^{2}
19 1+(1.61−0.930i)T+(0.5−0.866i)T2 1 + (1.61 - 0.930i)T + (0.5 - 0.866i)T^{2}
23 1+(0.826−0.563i)T2 1 + (0.826 - 0.563i)T^{2}
29 1+(−0.900+0.433i)T2 1 + (-0.900 + 0.433i)T^{2}
31 1+(−1.17−0.680i)T+(0.5+0.866i)T2 1 + (-1.17 - 0.680i)T + (0.5 + 0.866i)T^{2}
37 1+(−1.40+1.29i)T+(0.0747−0.997i)T2 1 + (-1.40 + 1.29i)T + (0.0747 - 0.997i)T^{2}
41 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
43 1+(1.78+0.858i)T+(0.623+0.781i)T2 1 + (1.78 + 0.858i)T + (0.623 + 0.781i)T^{2}
47 1+(0.733+0.680i)T2 1 + (0.733 + 0.680i)T^{2}
53 1+(0.0747+0.997i)T2 1 + (0.0747 + 0.997i)T^{2}
59 1+(−0.365+0.930i)T2 1 + (-0.365 + 0.930i)T^{2}
61 1+(−1.32−1.42i)T+(−0.0747+0.997i)T2 1 + (-1.32 - 1.42i)T + (-0.0747 + 0.997i)T^{2}
67 1+(0.826−1.43i)T+(−0.5−0.866i)T2 1 + (0.826 - 1.43i)T + (-0.5 - 0.866i)T^{2}
71 1+(−0.900−0.433i)T2 1 + (-0.900 - 0.433i)T^{2}
73 1+(−0.548+0.215i)T+(0.733−0.680i)T2 1 + (-0.548 + 0.215i)T + (0.733 - 0.680i)T^{2}
79 1+(−0.733−1.26i)T+(−0.5+0.866i)T2 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2}
83 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
89 1+(−0.955−0.294i)T2 1 + (-0.955 - 0.294i)T^{2}
97 1+1.56iT−T2 1 + 1.56iT - 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

−9.714490521548968212166206024888, −8.913597174780715938940791415255, −8.326852409993621433550167241234, −7.21204769002994785989324705132, −6.58917569887027356563718296683, −5.70969505752996765975276929971, −4.81488161385498421540891329398, −3.95875732969036900714205734194, −2.66999265226807694274770003678, −1.92719683531556375317729741188, 0.54408882113953902598296067263, 2.36801178250083128416213148718, 3.18421111788138313269918125658, 4.49825228991454747589270588122, 4.88491363054085725762209691712, 6.31599121857812702925562037982, 6.73719411044172293462028141027, 7.84914540446961035808838310766, 8.206417038410405124296802332101, 9.461734841124064988551166785497

Graph of the ZZ-function along the critical line