↑

Properties

Label 2-7400-1.1-c1-0-157
Degree 22
Conductor 74007400
Sign −1-1
Analytic cond. 59.089259.0892
Root an. cond. 7.686957.68695
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.60·3-s − 1.83·7-s + 3.77·9-s − 2.57·11-s − 1.16·13-s − 4.37·17-s + 8.21·19-s − 4.77·21-s − 5.87·23-s + 2.02·27-s + 2.45·29-s + 1.96·31-s − 6.70·33-s + 37-s − 3.04·39-s − 4.40·41-s − 10.9·43-s − 2.19·47-s − 3.63·49-s − 11.3·51-s + 6.44·53-s + 21.3·57-s − 5.31·59-s − 1.92·61-s − 6.93·63-s − 9.58·67-s − 15.2·69-s + ⋯
L(s)  = 1  + 1.50·3-s − 0.693·7-s + 1.25·9-s − 0.776·11-s − 0.324·13-s − 1.06·17-s + 1.88·19-s − 1.04·21-s − 1.22·23-s + 0.390·27-s + 0.455·29-s + 0.353·31-s − 1.16·33-s + 0.164·37-s − 0.487·39-s − 0.688·41-s − 1.66·43-s − 0.319·47-s − 0.519·49-s − 1.59·51-s + 0.884·53-s + 2.83·57-s − 0.691·59-s − 0.245·61-s − 0.873·63-s − 1.17·67-s − 1.84·69-s + ⋯

Functional equation

Λ(s)=(7400s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(7400s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 74007400    =    23⋅52⋅372^{3} \cdot 5^{2} \cdot 37
Sign: −1-1
Analytic conductor: 59.089259.0892
Root analytic conductor: 7.686957.68695
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 7400, ( :1/2), −1)(2,\ 7400,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad2 1 1
5 1 1
37 1−T 1 - T
good3 1−2.60T+3T2 1 - 2.60T + 3T^{2}
7 1+1.83T+7T2 1 + 1.83T + 7T^{2}
11 1+2.57T+11T2 1 + 2.57T + 11T^{2}
13 1+1.16T+13T2 1 + 1.16T + 13T^{2}
17 1+4.37T+17T2 1 + 4.37T + 17T^{2}
19 1−8.21T+19T2 1 - 8.21T + 19T^{2}
23 1+5.87T+23T2 1 + 5.87T + 23T^{2}
29 1−2.45T+29T2 1 - 2.45T + 29T^{2}
31 1−1.96T+31T2 1 - 1.96T + 31T^{2}
41 1+4.40T+41T2 1 + 4.40T + 41T^{2}
43 1+10.9T+43T2 1 + 10.9T + 43T^{2}
47 1+2.19T+47T2 1 + 2.19T + 47T^{2}
53 1−6.44T+53T2 1 - 6.44T + 53T^{2}
59 1+5.31T+59T2 1 + 5.31T + 59T^{2}
61 1+1.92T+61T2 1 + 1.92T + 61T^{2}
67 1+9.58T+67T2 1 + 9.58T + 67T^{2}
71 1−3.34T+71T2 1 - 3.34T + 71T^{2}
73 1+12.2T+73T2 1 + 12.2T + 73T^{2}
79 1−7.84T+79T2 1 - 7.84T + 79T^{2}
83 1−5.94T+83T2 1 - 5.94T + 83T^{2}
89 1+4.71T+89T2 1 + 4.71T + 89T^{2}
97 1−4.14T+97T2 1 - 4.14T + 97T^{2}
show more
show less
   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

−7.77380438221412235634322403413, −7.02392613848248657859546204110, −6.32968153020519332969779350118, −5.35354250813643235740755872156, −4.59087320496417846686539323993, −3.66248235735476381116198262435, −3.06085589887160388742016133900, −2.49253999766850301263740328006, −1.57425735451752144925734281963, 0, 1.57425735451752144925734281963, 2.49253999766850301263740328006, 3.06085589887160388742016133900, 3.66248235735476381116198262435, 4.59087320496417846686539323993, 5.35354250813643235740755872156, 6.32968153020519332969779350118, 7.02392613848248657859546204110, 7.77380438221412235634322403413

Graph of the ZZ-function along the critical line