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Properties

Label 2-2175-1.1-c1-0-14
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  + 0.510·2-s + 3-s − 1.73·4-s + 0.510·6-s − 4.82·7-s − 1.90·8-s + 9-s + 4.88·11-s − 1.73·12-s − 4.59·13-s − 2.46·14-s + 2.50·16-s − 6.50·17-s + 0.510·18-s + 3.09·19-s − 4.82·21-s + 2.49·22-s + 5.62·23-s − 1.90·24-s − 2.34·26-s + 27-s + 8.38·28-s + 29-s + 9.24·31-s + 5.09·32-s + 4.88·33-s − 3.32·34-s + ⋯
L(s)  = 1  + 0.361·2-s + 0.577·3-s − 0.869·4-s + 0.208·6-s − 1.82·7-s − 0.675·8-s + 0.333·9-s + 1.47·11-s − 0.502·12-s − 1.27·13-s − 0.658·14-s + 0.625·16-s − 1.57·17-s + 0.120·18-s + 0.709·19-s − 1.05·21-s + 0.531·22-s + 1.17·23-s − 0.389·24-s − 0.460·26-s + 0.192·27-s + 1.58·28-s + 0.185·29-s + 1.66·31-s + 0.901·32-s + 0.850·33-s − 0.570·34-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 1.5050921211.505092121
L(12)L(\frac12) ≈\approx 1.5050921211.505092121
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 1−0.510T+2T2 1 - 0.510T + 2T^{2}
7 1+4.82T+7T2 1 + 4.82T + 7T^{2}
11 1−4.88T+11T2 1 - 4.88T + 11T^{2}
13 1+4.59T+13T2 1 + 4.59T + 13T^{2}
17 1+6.50T+17T2 1 + 6.50T + 17T^{2}
19 1−3.09T+19T2 1 - 3.09T + 19T^{2}
23 1−5.62T+23T2 1 - 5.62T + 23T^{2}
31 1−9.24T+31T2 1 - 9.24T + 31T^{2}
37 1−11.1T+37T2 1 - 11.1T + 37T^{2}
41 1+2.84T+41T2 1 + 2.84T + 41T^{2}
43 1−4.58T+43T2 1 - 4.58T + 43T^{2}
47 1+3.62T+47T2 1 + 3.62T + 47T^{2}
53 1−0.967T+53T2 1 - 0.967T + 53T^{2}
59 1+0.298T+59T2 1 + 0.298T + 59T^{2}
61 1+0.786T+61T2 1 + 0.786T + 61T^{2}
67 1+4.86T+67T2 1 + 4.86T + 67T^{2}
71 1−0.741T+71T2 1 - 0.741T + 71T^{2}
73 1+5.52T+73T2 1 + 5.52T + 73T^{2}
79 1+2.96T+79T2 1 + 2.96T + 79T^{2}
83 1−13.6T+83T2 1 - 13.6T + 83T^{2}
89 1−3.67T+89T2 1 - 3.67T + 89T^{2}
97 1−2.87T+97T2 1 - 2.87T + 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

−9.234993594825525624364079927886, −8.604785510102191739950001517031, −7.37885169349480101593700280084, −6.62592425445843005796650472799, −6.10449477130991014213560895993, −4.76559855400695039982176234287, −4.19853072230293320891625549958, −3.25083143224501669743669053700, −2.61819624975453197322453712648, −0.73047933133623116124195629582, 0.73047933133623116124195629582, 2.61819624975453197322453712648, 3.25083143224501669743669053700, 4.19853072230293320891625549958, 4.76559855400695039982176234287, 6.10449477130991014213560895993, 6.62592425445843005796650472799, 7.37885169349480101593700280084, 8.604785510102191739950001517031, 9.234993594825525624364079927886

Graph of the ZZ-function along the critical line