↑

Properties

Label 2-232050-1.1-c1-0-123
Degree 22
Conductor 232050232050
Sign −1-1
Analytic cond. 1852.921852.92
Root an. cond. 43.045643.0456
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s − 4·11-s − 12-s − 13-s − 14-s + 16-s + 17-s + 18-s + 4·19-s + 21-s − 4·22-s + 8·23-s − 24-s − 26-s − 27-s − 28-s + 2·29-s − 8·31-s + 32-s + 4·33-s + 34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.218·21-s − 0.852·22-s + 1.66·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s − 1.43·31-s + 0.176·32-s + 0.696·33-s + 0.171·34-s + ⋯

Functional equation

Λ(s)=(232050s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 232050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(232050s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 232050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 232050232050    =    2⋅3⋅52⋅7⋅13⋅172 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13 \cdot 17
Sign: −1-1
Analytic conductor: 1852.921852.92
Root analytic conductor: 43.045643.0456
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 232050, ( :1/2), −1)(2,\ 232050,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{2}) 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
3 1+T 1 + T
5 1 1
7 1+T 1 + T
13 1+T 1 + T
17 1−T 1 - T
good11 1+4T+pT2 1 + 4 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1−8T+pT2 1 - 8 T + p T^{2}
71 1+4T+pT2 1 + 4 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+10T+pT2 1 + 10 T + p 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

−13.17421097541446, −12.69551024867424, −12.27865996923072, −11.90546070158911, −11.30037489690081, −10.79092708264778, −10.61382122476080, −10.02662097040330, −9.486526366430134, −9.022840712464805, −8.433821894106191, −7.643642040718000, −7.455363972156816, −6.885993461953966, −6.540485788488025, −5.631160018991992, −5.452515991199704, −5.131260473615614, −4.515306795893990, −3.840846215033406, −3.328516675156986, −2.765363418168176, −2.327937497961750, −1.447757998980346, −0.8100912233184165, 0, 0.8100912233184165, 1.447757998980346, 2.327937497961750, 2.765363418168176, 3.328516675156986, 3.840846215033406, 4.515306795893990, 5.131260473615614, 5.452515991199704, 5.631160018991992, 6.540485788488025, 6.885993461953966, 7.455363972156816, 7.643642040718000, 8.433821894106191, 9.022840712464805, 9.486526366430134, 10.02662097040330, 10.61382122476080, 10.79092708264778, 11.30037489690081, 11.90546070158911, 12.27865996923072, 12.69551024867424, 13.17421097541446

Graph of the ZZ-function along the critical line