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Properties

Label 2-1539-171.131-c0-0-0
Degree 22
Conductor 15391539
Sign 0.630+0.776i0.630 + 0.776i
Analytic cond. 0.7680610.768061
Root an. cond. 0.8763900.876390
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  + (0.766 − 0.642i)4-s + (0.939 − 1.62i)7-s + (0.266 + 1.50i)13-s + (0.173 − 0.984i)16-s + (−0.5 + 0.866i)19-s + (−0.939 − 0.342i)25-s + (−0.326 − 1.85i)28-s + 0.347·31-s − 1.87·37-s + (0.266 + 0.223i)43-s + (−1.26 − 2.19i)49-s + (1.17 + 0.984i)52-s + (0.266 + 1.50i)61-s + (−0.500 − 0.866i)64-s + (1.76 + 0.642i)67-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)4-s + (0.939 − 1.62i)7-s + (0.266 + 1.50i)13-s + (0.173 − 0.984i)16-s + (−0.5 + 0.866i)19-s + (−0.939 − 0.342i)25-s + (−0.326 − 1.85i)28-s + 0.347·31-s − 1.87·37-s + (0.266 + 0.223i)43-s + (−1.26 − 2.19i)49-s + (1.17 + 0.984i)52-s + (0.266 + 1.50i)61-s + (−0.500 − 0.866i)64-s + (1.76 + 0.642i)67-s + ⋯

Functional equation

Λ(s)=(1539s/2 ΓC(s) L(s)=((0.630+0.776i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1539s/2 ΓC(s) L(s)=((0.630+0.776i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15391539    =    34⋅193^{4} \cdot 19
Sign: 0.630+0.776i0.630 + 0.776i
Analytic conductor: 0.7680610.768061
Root analytic conductor: 0.8763900.876390
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1539(701,⋅)\chi_{1539} (701, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1539, ( :0), 0.630+0.776i)(2,\ 1539,\ (\ :0),\ 0.630 + 0.776i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4298608291.429860829
L(12)L(\frac12) ≈\approx 1.4298608291.429860829
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 1
19 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good2 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
5 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
7 1+(−0.939+1.62i)T+(−0.5−0.866i)T2 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2}
11 1−T2 1 - T^{2}
13 1+(−0.266−1.50i)T+(−0.939+0.342i)T2 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2}
17 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
23 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
29 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
31 1−0.347T+T2 1 - 0.347T + T^{2}
37 1+1.87T+T2 1 + 1.87T + T^{2}
41 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
43 1+(−0.266−0.223i)T+(0.173+0.984i)T2 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2}
47 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
53 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
59 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
61 1+(−0.266−1.50i)T+(−0.939+0.342i)T2 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2}
67 1+(−1.76−0.642i)T+(0.766+0.642i)T2 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2}
71 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
73 1+(−0.266−0.223i)T+(0.173+0.984i)T2 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2}
79 1+(−0.0603+0.342i)T+(−0.939−0.342i)T2 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2}
83 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
89 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
97 1+(1.87−0.684i)T+(0.766−0.642i)T2 1 + (1.87 - 0.684i)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

−9.839036635875506689744343577677, −8.650382044158021148447352052988, −7.79453633842501404835591115524, −7.03049023070501881955062667128, −6.50603252113923799294352320884, −5.42865492090309232827935010196, −4.40047188461384447810043274469, −3.76338598162392126990509651957, −2.07289366321321238224563888294, −1.32029729186612737803445188890, 1.86958807628796803968049289906, 2.65381813564669162668896695511, 3.57697497779721537029569596983, 4.99508638760062759611069931061, 5.63030651034592048352119998258, 6.47925228368821255103646689308, 7.51927962535623997856574907179, 8.305065138818329809778676915784, 8.607771762043015982129751714150, 9.692901482578110050797919721123

Graph of the ZZ-function along the critical line