↑

Properties

Label 2-644-644.643-c0-0-2
Degree 22
Conductor 644644
Sign 11
Analytic cond. 0.3213970.321397
Root an. cond. 0.5669190.566919
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.41·3-s + 4-s − 1.41·5-s − 1.41·6-s + 7-s − 8-s + 1.00·9-s + 1.41·10-s + 1.41·12-s − 14-s − 2.00·15-s + 16-s + 1.41·17-s − 1.00·18-s − 1.41·20-s + 1.41·21-s + 23-s − 1.41·24-s + 1.00·25-s + 28-s + 2.00·30-s − 1.41·31-s − 32-s − 1.41·34-s − 1.41·35-s + 1.00·36-s + ⋯
L(s)  = 1  − 2-s + 1.41·3-s + 4-s − 1.41·5-s − 1.41·6-s + 7-s − 8-s + 1.00·9-s + 1.41·10-s + 1.41·12-s − 14-s − 2.00·15-s + 16-s + 1.41·17-s − 1.00·18-s − 1.41·20-s + 1.41·21-s + 23-s − 1.41·24-s + 1.00·25-s + 28-s + 2.00·30-s − 1.41·31-s − 32-s − 1.41·34-s − 1.41·35-s + 1.00·36-s + ⋯

Functional equation

Λ(s)=(644s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(644s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 644644    =    22⋅7⋅232^{2} \cdot 7 \cdot 23
Sign: 11
Analytic conductor: 0.3213970.321397
Root analytic conductor: 0.5669190.566919
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ644(643,⋅)\chi_{644} (643, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 644, ( :0), 1)(2,\ 644,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.85584057300.8558405730
L(12)L(\frac12) ≈\approx 0.85584057300.8558405730
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+T 1 + T
7 1−T 1 - T
23 1−T 1 - T
good3 1−1.41T+T2 1 - 1.41T + T^{2}
5 1+1.41T+T2 1 + 1.41T + T^{2}
11 1+T2 1 + T^{2}
13 1−T2 1 - T^{2}
17 1−1.41T+T2 1 - 1.41T + T^{2}
19 1−T2 1 - T^{2}
29 1+T2 1 + T^{2}
31 1+1.41T+T2 1 + 1.41T + T^{2}
37 1−T2 1 - T^{2}
41 1−T2 1 - T^{2}
43 1+2T+T2 1 + 2T + T^{2}
47 1+1.41T+T2 1 + 1.41T + T^{2}
53 1−T2 1 - T^{2}
59 1−1.41T+T2 1 - 1.41T + T^{2}
61 1+1.41T+T2 1 + 1.41T + T^{2}
67 1+T2 1 + T^{2}
71 1−T2 1 - T^{2}
73 1−T2 1 - T^{2}
79 1+T2 1 + T^{2}
83 1−T2 1 - T^{2}
89 1−1.41T+T2 1 - 1.41T + T^{2}
97 1+1.41T+T2 1 + 1.41T + T^{2}
show more
show less
   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

−10.71151265954247160824932584218, −9.659033938005076050156005878228, −8.786175260830488425875697771192, −8.121857943294006454824228319259, −7.73792359421508395715617726156, −6.98784542127546217791065901538, −5.22167820167357457854019983097, −3.76390930507839525578237207628, −3.03562512031864936252004995522, −1.58755983396228822870147661188, 1.58755983396228822870147661188, 3.03562512031864936252004995522, 3.76390930507839525578237207628, 5.22167820167357457854019983097, 6.98784542127546217791065901538, 7.73792359421508395715617726156, 8.121857943294006454824228319259, 8.786175260830488425875697771192, 9.659033938005076050156005878228, 10.71151265954247160824932584218

Graph of the ZZ-function along the critical line