↑

Properties

Label 2-2175-87.86-c0-0-1
Degree 22
Conductor 21752175
Sign 11
Analytic cond. 1.085461.08546
Root an. cond. 1.041851.04185
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  − 0.347·2-s − 3-s − 0.879·4-s + 0.347·6-s − 1.87·7-s + 0.652·8-s + 9-s − 1.53·11-s + 0.879·12-s + 0.347·13-s + 0.652·14-s + 0.652·16-s − 1.53·17-s − 0.347·18-s + 1.87·21-s + 0.532·22-s − 0.652·24-s − 0.120·26-s − 27-s + 1.65·28-s − 29-s − 0.879·32-s + 1.53·33-s + 0.532·34-s − 0.879·36-s − 0.347·39-s + 41-s + ⋯
L(s)  = 1  − 0.347·2-s − 3-s − 0.879·4-s + 0.347·6-s − 1.87·7-s + 0.652·8-s + 9-s − 1.53·11-s + 0.879·12-s + 0.347·13-s + 0.652·14-s + 0.652·16-s − 1.53·17-s − 0.347·18-s + 1.87·21-s + 0.532·22-s − 0.652·24-s − 0.120·26-s − 27-s + 1.65·28-s − 29-s − 0.879·32-s + 1.53·33-s + 0.532·34-s − 0.879·36-s − 0.347·39-s + 41-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 21752175    =    3⋅52⋅293 \cdot 5^{2} \cdot 29
Sign: 11
Analytic conductor: 1.085461.08546
Root analytic conductor: 1.041851.04185
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2175(1826,⋅)\chi_{2175} (1826, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2175, ( :0), 1)(2,\ 2175,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.23945352730.2394535273
L(12)L(\frac12) ≈\approx 0.23945352730.2394535273
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+T 1 + T
5 1 1
29 1+T 1 + T
good2 1+0.347T+T2 1 + 0.347T + T^{2}
7 1+1.87T+T2 1 + 1.87T + T^{2}
11 1+1.53T+T2 1 + 1.53T + T^{2}
13 1−0.347T+T2 1 - 0.347T + T^{2}
17 1+1.53T+T2 1 + 1.53T + T^{2}
19 1−T2 1 - T^{2}
23 1−T2 1 - T^{2}
31 1−T2 1 - T^{2}
37 1−T2 1 - T^{2}
41 1−T+T2 1 - T + T^{2}
43 1−T2 1 - T^{2}
47 1−1.87T+T2 1 - 1.87T + T^{2}
53 1−T2 1 - T^{2}
59 1−T2 1 - T^{2}
61 1−T2 1 - T^{2}
67 1−1.53T+T2 1 - 1.53T + 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+0.347T+T2 1 + 0.347T + T^{2}
97 1−T2 1 - 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

−9.396118514439231217003398098903, −8.685899058452189065267677146579, −7.58483552922305159556451678421, −6.92166442159257411947785103629, −6.00700754698867846014773369746, −5.44725657725167988838580996213, −4.43708997874958087758069894207, −3.66809801714371974600384617121, −2.41317590431588517460914701473, −0.49730740771534572588371024792, 0.49730740771534572588371024792, 2.41317590431588517460914701473, 3.66809801714371974600384617121, 4.43708997874958087758069894207, 5.44725657725167988838580996213, 6.00700754698867846014773369746, 6.92166442159257411947785103629, 7.58483552922305159556451678421, 8.685899058452189065267677146579, 9.396118514439231217003398098903

Graph of the ZZ-function along the critical line