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Properties

Label 2-2175-87.86-c0-0-14
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  + 1.87·2-s − 3-s + 2.53·4-s − 1.87·6-s + 1.53·7-s + 2.87·8-s + 9-s − 0.347·11-s − 2.53·12-s − 1.87·13-s + 2.87·14-s + 2.87·16-s − 0.347·17-s + 1.87·18-s − 1.53·21-s − 0.652·22-s − 2.87·24-s − 3.53·26-s − 27-s + 3.87·28-s − 29-s + 2.53·32-s + 0.347·33-s − 0.652·34-s + 2.53·36-s + 1.87·39-s + 41-s + ⋯
L(s)  = 1  + 1.87·2-s − 3-s + 2.53·4-s − 1.87·6-s + 1.53·7-s + 2.87·8-s + 9-s − 0.347·11-s − 2.53·12-s − 1.87·13-s + 2.87·14-s + 2.87·16-s − 0.347·17-s + 1.87·18-s − 1.53·21-s − 0.652·22-s − 2.87·24-s − 3.53·26-s − 27-s + 3.87·28-s − 29-s + 2.53·32-s + 0.347·33-s − 0.652·34-s + 2.53·36-s + 1.87·39-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 2.9574012972.957401297
L(12)L(\frac12) ≈\approx 2.9574012972.957401297
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 1−1.87T+T2 1 - 1.87T + T^{2}
7 1−1.53T+T2 1 - 1.53T + T^{2}
11 1+0.347T+T2 1 + 0.347T + T^{2}
13 1+1.87T+T2 1 + 1.87T + T^{2}
17 1+0.347T+T2 1 + 0.347T + 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+1.53T+T2 1 + 1.53T + T^{2}
53 1−T2 1 - T^{2}
59 1−T2 1 - T^{2}
61 1−T2 1 - T^{2}
67 1−0.347T+T2 1 - 0.347T + 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 1−1.87T+T2 1 - 1.87T + 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

−9.523391246911277236197277394714, −7.898304037184150965892262877441, −7.43982266164931105211698921052, −6.68578315958118128747274737475, −5.72934784548585606706766407868, −4.99712873200408410429398761002, −4.80202232634207885304732900559, −3.92581744283433777952756655802, −2.51610797159669187321861614201, −1.73070337497445164535221643354, 1.73070337497445164535221643354, 2.51610797159669187321861614201, 3.92581744283433777952756655802, 4.80202232634207885304732900559, 4.99712873200408410429398761002, 5.72934784548585606706766407868, 6.68578315958118128747274737475, 7.43982266164931105211698921052, 7.898304037184150965892262877441, 9.523391246911277236197277394714

Graph of the ZZ-function along the critical line