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Properties

Label 2-336-1.1-c7-0-19
Degree 22
Conductor 336336
Sign −1-1
Analytic cond. 104.961104.961
Root an. cond. 10.245010.2450
Motivic weight 77
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 27·3-s − 470·5-s + 343·7-s + 729·9-s − 1.84e3·11-s + 758·13-s + 1.26e4·15-s − 2.80e4·17-s + 4.59e4·19-s − 9.26e3·21-s − 6.23e4·23-s + 1.42e5·25-s − 1.96e4·27-s + 7.63e4·29-s + 1.31e5·31-s + 4.96e4·33-s − 1.61e5·35-s + 3.98e5·37-s − 2.04e4·39-s + 7.13e5·41-s + 2.87e4·43-s − 3.42e5·45-s − 6.65e4·47-s + 1.17e5·49-s + 7.57e5·51-s + 1.29e6·53-s + 8.64e5·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.68·5-s + 0.377·7-s + 1/3·9-s − 0.416·11-s + 0.0956·13-s + 0.970·15-s − 1.38·17-s + 1.53·19-s − 0.218·21-s − 1.06·23-s + 1.82·25-s − 0.192·27-s + 0.581·29-s + 0.793·31-s + 0.240·33-s − 0.635·35-s + 1.29·37-s − 0.0552·39-s + 1.61·41-s + 0.0551·43-s − 0.560·45-s − 0.0934·47-s + 1/7·49-s + 0.800·51-s + 1.19·53-s + 0.700·55-s + ⋯

Functional equation

Λ(s)=(336s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(336s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 336336    =    24⋅3⋅72^{4} \cdot 3 \cdot 7
Sign: −1-1
Analytic conductor: 104.961104.961
Root analytic conductor: 10.245010.2450
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 336, ( :7/2), −1)(2,\ 336,\ (\ :7/2),\ -1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
L(92)L(\frac{9}{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+p3T 1 + p^{3} T
7 1−p3T 1 - p^{3} T
good5 1+94pT+p7T2 1 + 94 p T + p^{7} T^{2}
11 1+1840T+p7T2 1 + 1840 T + p^{7} T^{2}
13 1−758T+p7T2 1 - 758 T + p^{7} T^{2}
17 1+28074T+p7T2 1 + 28074 T + p^{7} T^{2}
19 1−45964T+p7T2 1 - 45964 T + p^{7} T^{2}
23 1+62388T+p7T2 1 + 62388 T + p^{7} T^{2}
29 1−76350T+p7T2 1 - 76350 T + p^{7} T^{2}
31 1−131608T+p7T2 1 - 131608 T + p^{7} T^{2}
37 1−398302T+p7T2 1 - 398302 T + p^{7} T^{2}
41 1−713878T+p7T2 1 - 713878 T + p^{7} T^{2}
43 1−28732T+p7T2 1 - 28732 T + p^{7} T^{2}
47 1+66536T+p7T2 1 + 66536 T + p^{7} T^{2}
53 1−1298190T+p7T2 1 - 1298190 T + p^{7} T^{2}
59 1+1391148T+p7T2 1 + 1391148 T + p^{7} T^{2}
61 1−176718T+p7T2 1 - 176718 T + p^{7} T^{2}
67 1−1776316T+p7T2 1 - 1776316 T + p^{7} T^{2}
71 1+4271948T+p7T2 1 + 4271948 T + p^{7} T^{2}
73 1+4333742T+p7T2 1 + 4333742 T + p^{7} T^{2}
79 1+5771608T+p7T2 1 + 5771608 T + p^{7} T^{2}
83 1+2231596T+p7T2 1 + 2231596 T + p^{7} T^{2}
89 1+2844858T+p7T2 1 + 2844858 T + p^{7} T^{2}
97 1−10995690T+p7T2 1 - 10995690 T + p^{7} T^{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

−10.06751396284755050487590056083, −8.766926663063306060938156321591, −7.83711715665402621355991520630, −7.23367984794848062668579830388, −5.98536170928427807705861703007, −4.68729106515621719227420897389, −4.07687360952368824765778892869, −2.73079204744218209092662342045, −0.983822571355788850850709103626, 0, 0.983822571355788850850709103626, 2.73079204744218209092662342045, 4.07687360952368824765778892869, 4.68729106515621719227420897389, 5.98536170928427807705861703007, 7.23367984794848062668579830388, 7.83711715665402621355991520630, 8.766926663063306060938156321591, 10.06751396284755050487590056083

Graph of the ZZ-function along the critical line