↑

Properties

Label 2-2175-1.1-c1-0-40
Degree 22
Conductor 21752175
Sign −1-1
Analytic cond. 17.367417.3674
Root an. cond. 4.167424.16742
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.754·2-s − 3-s − 1.43·4-s + 0.754·6-s − 4.18·7-s + 2.58·8-s + 9-s + 0.596·11-s + 1.43·12-s + 2.18·13-s + 3.15·14-s + 0.908·16-s − 2.81·17-s − 0.754·18-s + 0.528·19-s + 4.18·21-s − 0.450·22-s − 0.590·23-s − 2.58·24-s − 1.64·26-s − 27-s + 5.98·28-s + 29-s + 8.02·31-s − 5.86·32-s − 0.596·33-s + 2.12·34-s + ⋯
L(s)  = 1  − 0.533·2-s − 0.577·3-s − 0.715·4-s + 0.308·6-s − 1.58·7-s + 0.915·8-s + 0.333·9-s + 0.179·11-s + 0.413·12-s + 0.606·13-s + 0.843·14-s + 0.227·16-s − 0.682·17-s − 0.177·18-s + 0.121·19-s + 0.913·21-s − 0.0960·22-s − 0.123·23-s − 0.528·24-s − 0.323·26-s − 0.192·27-s + 1.13·28-s + 0.185·29-s + 1.44·31-s − 1.03·32-s − 0.103·33-s + 0.363·34-s + ⋯

Functional equation

Λ(s)=(2175s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2175s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21752175    =    3⋅52⋅293 \cdot 5^{2} \cdot 29
Sign: −1-1
Analytic conductor: 17.367417.3674
Root analytic conductor: 4.167424.16742
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2175, ( :1/2), −1)(2,\ 2175,\ (\ :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)
bad3 1+T 1 + T
5 1 1
29 1−T 1 - T
good2 1+0.754T+2T2 1 + 0.754T + 2T^{2}
7 1+4.18T+7T2 1 + 4.18T + 7T^{2}
11 1−0.596T+11T2 1 - 0.596T + 11T^{2}
13 1−2.18T+13T2 1 - 2.18T + 13T^{2}
17 1+2.81T+17T2 1 + 2.81T + 17T^{2}
19 1−0.528T+19T2 1 - 0.528T + 19T^{2}
23 1+0.590T+23T2 1 + 0.590T + 23T^{2}
31 1−8.02T+31T2 1 - 8.02T + 31T^{2}
37 1−2.52T+37T2 1 - 2.52T + 37T^{2}
41 1−1.57T+41T2 1 - 1.57T + 41T^{2}
43 1−6.98T+43T2 1 - 6.98T + 43T^{2}
47 1−2.57T+47T2 1 - 2.57T + 47T^{2}
53 1+10.3T+53T2 1 + 10.3T + 53T^{2}
59 1−13.9T+59T2 1 - 13.9T + 59T^{2}
61 1+11.1T+61T2 1 + 11.1T + 61T^{2}
67 1+0.614T+67T2 1 + 0.614T + 67T^{2}
71 1−9.28T+71T2 1 - 9.28T + 71T^{2}
73 1+9.59T+73T2 1 + 9.59T + 73T^{2}
79 1+10.3T+79T2 1 + 10.3T + 79T^{2}
83 1+3.92T+83T2 1 + 3.92T + 83T^{2}
89 1+12.3T+89T2 1 + 12.3T + 89T^{2}
97 1+3.21T+97T2 1 + 3.21T + 97T^{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

−8.859526112336300134003478691936, −8.032906487908560067729109043697, −7.04030914180551224372856395710, −6.35280605799936789759369877241, −5.67977854501064474690570106947, −4.53246181097811265902315628208, −3.86896209949477103890157631348, −2.77032007654021612627617461231, −1.11834923788114409681819649097, 0, 1.11834923788114409681819649097, 2.77032007654021612627617461231, 3.86896209949477103890157631348, 4.53246181097811265902315628208, 5.67977854501064474690570106947, 6.35280605799936789759369877241, 7.04030914180551224372856395710, 8.032906487908560067729109043697, 8.859526112336300134003478691936

Graph of the ZZ-function along the critical line