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Properties

Label 2-1680-1.1-c3-0-40
Degree 22
Conductor 16801680
Sign −1-1
Analytic cond. 99.123299.1232
Root an. cond. 9.956069.95606
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 5·5-s − 7·7-s + 9·9-s − 37.4·11-s + 29.0·13-s + 15·15-s + 58.4·17-s + 54.5·19-s + 21·21-s − 161.·23-s + 25·25-s − 27·27-s + 137.·29-s − 154.·31-s + 112.·33-s + 35·35-s − 350.·37-s − 87.0·39-s + 353.·41-s + 518.·43-s − 45·45-s + 542.·47-s + 49·49-s − 175.·51-s + 305.·53-s + 187.·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s + 0.333·9-s − 1.02·11-s + 0.619·13-s + 0.258·15-s + 0.833·17-s + 0.659·19-s + 0.218·21-s − 1.46·23-s + 0.200·25-s − 0.192·27-s + 0.880·29-s − 0.896·31-s + 0.591·33-s + 0.169·35-s − 1.55·37-s − 0.357·39-s + 1.34·41-s + 1.83·43-s − 0.149·45-s + 1.68·47-s + 0.142·49-s − 0.481·51-s + 0.791·53-s + 0.458·55-s + ⋯

Functional equation

Λ(s)=(1680s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1680s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16801680    =    24⋅3⋅5⋅72^{4} \cdot 3 \cdot 5 \cdot 7
Sign: −1-1
Analytic conductor: 99.123299.1232
Root analytic conductor: 9.956069.95606
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1680, ( :3/2), −1)(2,\ 1680,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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
3 1+3T 1 + 3T
5 1+5T 1 + 5T
7 1+7T 1 + 7T
good11 1+37.4T+1.33e3T2 1 + 37.4T + 1.33e3T^{2}
13 1−29.0T+2.19e3T2 1 - 29.0T + 2.19e3T^{2}
17 1−58.4T+4.91e3T2 1 - 58.4T + 4.91e3T^{2}
19 1−54.5T+6.85e3T2 1 - 54.5T + 6.85e3T^{2}
23 1+161.T+1.21e4T2 1 + 161.T + 1.21e4T^{2}
29 1−137.T+2.43e4T2 1 - 137.T + 2.43e4T^{2}
31 1+154.T+2.97e4T2 1 + 154.T + 2.97e4T^{2}
37 1+350.T+5.06e4T2 1 + 350.T + 5.06e4T^{2}
41 1−353.T+6.89e4T2 1 - 353.T + 6.89e4T^{2}
43 1−518.T+7.95e4T2 1 - 518.T + 7.95e4T^{2}
47 1−542.T+1.03e5T2 1 - 542.T + 1.03e5T^{2}
53 1−305.T+1.48e5T2 1 - 305.T + 1.48e5T^{2}
59 1+14.6T+2.05e5T2 1 + 14.6T + 2.05e5T^{2}
61 1+171.T+2.26e5T2 1 + 171.T + 2.26e5T^{2}
67 1+551.T+3.00e5T2 1 + 551.T + 3.00e5T^{2}
71 1−120.T+3.57e5T2 1 - 120.T + 3.57e5T^{2}
73 1−284.T+3.89e5T2 1 - 284.T + 3.89e5T^{2}
79 1+941.T+4.93e5T2 1 + 941.T + 4.93e5T^{2}
83 1+377.T+5.71e5T2 1 + 377.T + 5.71e5T^{2}
89 1+677.T+7.04e5T2 1 + 677.T + 7.04e5T^{2}
97 1+1.22e3T+9.12e5T2 1 + 1.22e3T + 9.12e5T^{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

−8.531756647116448101446720585050, −7.65799538734073440975979290978, −7.15152862733662241905964892920, −5.88166212268994993932282941827, −5.58669225128133447694363234460, −4.38301460074768451552671671976, −3.56974716056325902453382840958, −2.49988520297808149484077279092, −1.07404457693876035493318885050, 0, 1.07404457693876035493318885050, 2.49988520297808149484077279092, 3.56974716056325902453382840958, 4.38301460074768451552671671976, 5.58669225128133447694363234460, 5.88166212268994993932282941827, 7.15152862733662241905964892920, 7.65799538734073440975979290978, 8.531756647116448101446720585050

Graph of the ZZ-function along the critical line