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Properties

Label 2-2175-87.86-c0-0-9
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 0.86574788250.8657478825
L(12)L(\frac12) ≈\approx 0.86574788250.8657478825
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

−9.177276752625871114300195073054, −8.408644787817091106536694094947, −7.991267099831890594699457123940, −7.31490326498810007029635375582, −6.80849427207487440746421475832, −5.33594900268624110175744969100, −4.33743472566190014343759443314, −2.83614782990033014927984572165, −2.13497346391147626444047201176, −1.25939835582413232369720352277, 1.25939835582413232369720352277, 2.13497346391147626444047201176, 2.83614782990033014927984572165, 4.33743472566190014343759443314, 5.33594900268624110175744969100, 6.80849427207487440746421475832, 7.31490326498810007029635375582, 7.991267099831890594699457123940, 8.408644787817091106536694094947, 9.177276752625871114300195073054

Graph of the ZZ-function along the critical line