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Properties

Label 2-2175-1.1-c1-0-25
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  − 2.66·2-s − 3-s + 5.09·4-s + 2.66·6-s + 0.0170·7-s − 8.24·8-s + 9-s + 1.70·11-s − 5.09·12-s + 4.69·13-s − 0.0453·14-s + 11.7·16-s + 3.91·17-s − 2.66·18-s + 6.02·19-s − 0.0170·21-s − 4.55·22-s − 4.82·23-s + 8.24·24-s − 12.5·26-s − 27-s + 0.0868·28-s − 29-s + 1.33·31-s − 14.8·32-s − 1.70·33-s − 10.4·34-s + ⋯
L(s)  = 1  − 1.88·2-s − 0.577·3-s + 2.54·4-s + 1.08·6-s + 0.00644·7-s − 2.91·8-s + 0.333·9-s + 0.515·11-s − 1.47·12-s + 1.30·13-s − 0.0121·14-s + 2.94·16-s + 0.949·17-s − 0.627·18-s + 1.38·19-s − 0.00371·21-s − 0.971·22-s − 1.00·23-s + 1.68·24-s − 2.45·26-s − 0.192·27-s + 0.0164·28-s − 0.185·29-s + 0.240·31-s − 2.62·32-s − 0.297·33-s − 1.78·34-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 0.72139636840.7213963684
L(12)L(\frac12) ≈\approx 0.72139636840.7213963684
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+2.66T+2T2 1 + 2.66T + 2T^{2}
7 1−0.0170T+7T2 1 - 0.0170T + 7T^{2}
11 1−1.70T+11T2 1 - 1.70T + 11T^{2}
13 1−4.69T+13T2 1 - 4.69T + 13T^{2}
17 1−3.91T+17T2 1 - 3.91T + 17T^{2}
19 1−6.02T+19T2 1 - 6.02T + 19T^{2}
23 1+4.82T+23T2 1 + 4.82T + 23T^{2}
31 1−1.33T+31T2 1 - 1.33T + 31T^{2}
37 1−8.18T+37T2 1 - 8.18T + 37T^{2}
41 1−8.36T+41T2 1 - 8.36T + 41T^{2}
43 1+3.72T+43T2 1 + 3.72T + 43T^{2}
47 1−5.36T+47T2 1 - 5.36T + 47T^{2}
53 1−7.21T+53T2 1 - 7.21T + 53T^{2}
59 1+8.65T+59T2 1 + 8.65T + 59T^{2}
61 1−5.72T+61T2 1 - 5.72T + 61T^{2}
67 1+3.87T+67T2 1 + 3.87T + 67T^{2}
71 1+4.32T+71T2 1 + 4.32T + 71T^{2}
73 1−1.78T+73T2 1 - 1.78T + 73T^{2}
79 1−0.233T+79T2 1 - 0.233T + 79T^{2}
83 1−6.15T+83T2 1 - 6.15T + 83T^{2}
89 1+12.4T+89T2 1 + 12.4T + 89T^{2}
97 1−19.2T+97T2 1 - 19.2T + 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

−9.304905015397407057984508886823, −8.218152379239932118776538293800, −7.78918940137861932611097369539, −6.94241326170045472754322790506, −6.12937338899109650037635792537, −5.60728342299222017101273006059, −3.98402738369101836133915587932, −2.90763899115558260939117157652, −1.53708590639601758346834133363, −0.834733188292331876571569817971, 0.834733188292331876571569817971, 1.53708590639601758346834133363, 2.90763899115558260939117157652, 3.98402738369101836133915587932, 5.60728342299222017101273006059, 6.12937338899109650037635792537, 6.94241326170045472754322790506, 7.78918940137861932611097369539, 8.218152379239932118776538293800, 9.304905015397407057984508886823

Graph of the ZZ-function along the critical line