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Properties

Label 2-2175-87.86-c0-0-6
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.53·2-s − 3-s + 1.34·4-s + 1.53·6-s + 0.347·7-s − 0.532·8-s + 9-s + 1.87·11-s − 1.34·12-s + 1.53·13-s − 0.532·14-s − 0.532·16-s + 1.87·17-s − 1.53·18-s − 0.347·21-s − 2.87·22-s + 0.532·24-s − 2.34·26-s − 27-s + 0.467·28-s − 29-s + 1.34·32-s − 1.87·33-s − 2.87·34-s + 1.34·36-s − 1.53·39-s + 41-s + ⋯
L(s)  = 1  − 1.53·2-s − 3-s + 1.34·4-s + 1.53·6-s + 0.347·7-s − 0.532·8-s + 9-s + 1.87·11-s − 1.34·12-s + 1.53·13-s − 0.532·14-s − 0.532·16-s + 1.87·17-s − 1.53·18-s − 0.347·21-s − 2.87·22-s + 0.532·24-s − 2.34·26-s − 27-s + 0.467·28-s − 29-s + 1.34·32-s − 1.87·33-s − 2.87·34-s + 1.34·36-s − 1.53·39-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 0.55310895430.5531089543
L(12)L(\frac12) ≈\approx 0.55310895430.5531089543
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+1.53T+T2 1 + 1.53T + T^{2}
7 1−0.347T+T2 1 - 0.347T + T^{2}
11 1−1.87T+T2 1 - 1.87T + T^{2}
13 1−1.53T+T2 1 - 1.53T + T^{2}
17 1−1.87T+T2 1 - 1.87T + 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+0.347T+T2 1 + 0.347T + T^{2}
53 1−T2 1 - T^{2}
59 1−T2 1 - T^{2}
61 1−T2 1 - T^{2}
67 1+1.87T+T2 1 + 1.87T + 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+1.53T+T2 1 + 1.53T + T^{2}
97 1−T2 1 - 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

−9.370231798785217174267696211616, −8.579943281937858846332289848704, −7.78568402852589979850302836332, −7.06299408771861551595904807137, −6.25909602560367724146302844124, −5.67830604541890026810108894041, −4.35847235339394158100138240385, −3.52757895222731274896147097187, −1.51489068005775694053626102262, −1.16696546040772382780751521796, 1.16696546040772382780751521796, 1.51489068005775694053626102262, 3.52757895222731274896147097187, 4.35847235339394158100138240385, 5.67830604541890026810108894041, 6.25909602560367724146302844124, 7.06299408771861551595904807137, 7.78568402852589979850302836332, 8.579943281937858846332289848704, 9.370231798785217174267696211616

Graph of the ZZ-function along the critical line