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Properties

Label 2-2793-2793.2105-c0-0-0
Degree 22
Conductor 27932793
Sign 0.922−0.385i0.922 - 0.385i
Analytic cond. 1.393881.39388
Root an. cond. 1.180631.18063
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  + (−0.980 − 0.198i)3-s + (−0.270 − 0.962i)4-s + (0.542 + 0.840i)7-s + (0.921 + 0.388i)9-s + (0.0747 + 0.997i)12-s + (0.636 − 0.0317i)13-s + (−0.853 + 0.521i)16-s + (0.222 + 0.974i)19-s + (−0.365 − 0.930i)21-s + (0.124 + 0.992i)25-s + (−0.826 − 0.563i)27-s + (0.661 − 0.749i)28-s − 1.93·31-s + (0.124 − 0.992i)36-s + (1.08 + 1.59i)37-s + ⋯
L(s)  = 1  + (−0.980 − 0.198i)3-s + (−0.270 − 0.962i)4-s + (0.542 + 0.840i)7-s + (0.921 + 0.388i)9-s + (0.0747 + 0.997i)12-s + (0.636 − 0.0317i)13-s + (−0.853 + 0.521i)16-s + (0.222 + 0.974i)19-s + (−0.365 − 0.930i)21-s + (0.124 + 0.992i)25-s + (−0.826 − 0.563i)27-s + (0.661 − 0.749i)28-s − 1.93·31-s + (0.124 − 0.992i)36-s + (1.08 + 1.59i)37-s + ⋯

Functional equation

Λ(s)=(2793s/2 ΓC(s) L(s)=((0.922−0.385i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2793s/2 ΓC(s) L(s)=((0.922−0.385i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27932793    =    3⋅72⋅193 \cdot 7^{2} \cdot 19
Sign: 0.922−0.385i0.922 - 0.385i
Analytic conductor: 1.393881.39388
Root analytic conductor: 1.180631.18063
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2793(2105,⋅)\chi_{2793} (2105, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2793, ( :0), 0.922−0.385i)(2,\ 2793,\ (\ :0),\ 0.922 - 0.385i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.83393622910.8339362291
L(12)L(\frac12) ≈\approx 0.83393622910.8339362291
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)
bad3 1+(0.980+0.198i)T 1 + (0.980 + 0.198i)T
7 1+(−0.542−0.840i)T 1 + (-0.542 - 0.840i)T
19 1+(−0.222−0.974i)T 1 + (-0.222 - 0.974i)T
good2 1+(0.270+0.962i)T2 1 + (0.270 + 0.962i)T^{2}
5 1+(−0.124−0.992i)T2 1 + (-0.124 - 0.992i)T^{2}
11 1+(−0.826−0.563i)T2 1 + (-0.826 - 0.563i)T^{2}
13 1+(−0.636+0.0317i)T+(0.995−0.0995i)T2 1 + (-0.636 + 0.0317i)T + (0.995 - 0.0995i)T^{2}
17 1+(−0.853−0.521i)T2 1 + (-0.853 - 0.521i)T^{2}
23 1+(0.0249+0.999i)T2 1 + (0.0249 + 0.999i)T^{2}
29 1+(0.980+0.198i)T2 1 + (0.980 + 0.198i)T^{2}
31 1+1.93T+T2 1 + 1.93T + T^{2}
37 1+(−1.08−1.59i)T+(−0.365+0.930i)T2 1 + (-1.08 - 1.59i)T + (-0.365 + 0.930i)T^{2}
41 1+(0.124+0.992i)T2 1 + (0.124 + 0.992i)T^{2}
43 1+(−0.0459−1.84i)T+(−0.998+0.0498i)T2 1 + (-0.0459 - 1.84i)T + (-0.998 + 0.0498i)T^{2}
47 1+(0.995−0.0995i)T2 1 + (0.995 - 0.0995i)T^{2}
53 1+(−0.853+0.521i)T2 1 + (-0.853 + 0.521i)T^{2}
59 1+(−0.542−0.840i)T2 1 + (-0.542 - 0.840i)T^{2}
61 1+(−0.0491+0.491i)T+(−0.980−0.198i)T2 1 + (-0.0491 + 0.491i)T + (-0.980 - 0.198i)T^{2}
67 1+(−1.21+1.45i)T+(−0.173−0.984i)T2 1 + (-1.21 + 1.45i)T + (-0.173 - 0.984i)T^{2}
71 1+(−0.661−0.749i)T2 1 + (-0.661 - 0.749i)T^{2}
73 1+(−1.27+0.653i)T+(0.583−0.811i)T2 1 + (-1.27 + 0.653i)T + (0.583 - 0.811i)T^{2}
79 1+(−0.623+1.71i)T+(−0.766−0.642i)T2 1 + (-0.623 + 1.71i)T + (-0.766 - 0.642i)T^{2}
83 1+(0.0747+0.997i)T2 1 + (0.0747 + 0.997i)T^{2}
89 1+(−0.270+0.962i)T2 1 + (-0.270 + 0.962i)T^{2}
97 1+(1.37+0.501i)T+(0.766+0.642i)T2 1 + (1.37 + 0.501i)T + (0.766 + 0.642i)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

−9.259383982642252335594656117193, −8.235376078627687899570051002470, −7.51905691990990773666212728411, −6.34915479228192493032173901219, −6.01591521895240884979290455733, −5.20371070758442315430413488461, −4.72904419198952320915193149711, −3.54013988646514979016592668277, −1.95500006417666435964194268237, −1.24868188878037167382302090498, 0.70132182927891253542825197811, 2.24310572280930145318567243242, 3.73694432512212714613567355278, 4.08261515862209721595071824226, 4.99877620191415591400316710017, 5.75233789968669131798530676574, 6.97194927915596395605240174877, 7.18232206404451035524621949586, 8.142160870330001531121584681027, 8.920563611207270525620761141383

Graph of the ZZ-function along the critical line