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Properties

Label 2-2175-1.1-c1-0-1
Degree 22
Conductor 21752175
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.57·2-s − 3-s + 4.64·4-s + 2.57·6-s − 4.69·7-s − 6.81·8-s + 9-s − 3.11·11-s − 4.64·12-s + 5.07·13-s + 12.1·14-s + 8.28·16-s − 1.40·17-s − 2.57·18-s − 3.76·19-s + 4.69·21-s + 8.03·22-s − 5.71·23-s + 6.81·24-s − 13.0·26-s − 27-s − 21.8·28-s + 29-s − 2.23·31-s − 7.73·32-s + 3.11·33-s + 3.60·34-s + ⋯
L(s)  = 1  − 1.82·2-s − 0.577·3-s + 2.32·4-s + 1.05·6-s − 1.77·7-s − 2.41·8-s + 0.333·9-s − 0.939·11-s − 1.34·12-s + 1.40·13-s + 3.23·14-s + 2.07·16-s − 0.339·17-s − 0.607·18-s − 0.863·19-s + 1.02·21-s + 1.71·22-s − 1.19·23-s + 1.39·24-s − 2.56·26-s − 0.192·27-s − 4.12·28-s + 0.185·29-s − 0.400·31-s − 1.36·32-s + 0.542·33-s + 0.619·34-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: 11
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: 00
Selberg data: (2, 2175, ( :1/2), 1)(2,\ 2175,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.19075598890.1907559889
L(12)L(\frac12) ≈\approx 0.19075598890.1907559889
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+2.57T+2T2 1 + 2.57T + 2T^{2}
7 1+4.69T+7T2 1 + 4.69T + 7T^{2}
11 1+3.11T+11T2 1 + 3.11T + 11T^{2}
13 1−5.07T+13T2 1 - 5.07T + 13T^{2}
17 1+1.40T+17T2 1 + 1.40T + 17T^{2}
19 1+3.76T+19T2 1 + 3.76T + 19T^{2}
23 1+5.71T+23T2 1 + 5.71T + 23T^{2}
31 1+2.23T+31T2 1 + 2.23T + 31T^{2}
37 1+5.79T+37T2 1 + 5.79T + 37T^{2}
41 1+10.6T+41T2 1 + 10.6T + 41T^{2}
43 1−8.89T+43T2 1 - 8.89T + 43T^{2}
47 1+3.62T+47T2 1 + 3.62T + 47T^{2}
53 1−0.948T+53T2 1 - 0.948T + 53T^{2}
59 1+8.53T+59T2 1 + 8.53T + 59T^{2}
61 1+6.21T+61T2 1 + 6.21T + 61T^{2}
67 1−13.9T+67T2 1 - 13.9T + 67T^{2}
71 1−5.88T+71T2 1 - 5.88T + 71T^{2}
73 1+7.08T+73T2 1 + 7.08T + 73T^{2}
79 1−7.31T+79T2 1 - 7.31T + 79T^{2}
83 1+13.6T+83T2 1 + 13.6T + 83T^{2}
89 1+7.98T+89T2 1 + 7.98T + 89T^{2}
97 1−13.8T+97T2 1 - 13.8T + 97T^{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

−9.084181927340036589792838591766, −8.464313309952598701659844387715, −7.67458491261962899226467640778, −6.65948518864007495867065960067, −6.39905280928136446165861319993, −5.58634779304436625833057777724, −3.90458636026905220803120738912, −2.92319217425504110822603056912, −1.80901281345292221085470937463, −0.36888289924507259664087467196, 0.36888289924507259664087467196, 1.80901281345292221085470937463, 2.92319217425504110822603056912, 3.90458636026905220803120738912, 5.58634779304436625833057777724, 6.39905280928136446165861319993, 6.65948518864007495867065960067, 7.67458491261962899226467640778, 8.464313309952598701659844387715, 9.084181927340036589792838591766

Graph of the ZZ-function along the critical line