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Properties

Label 2-338-1.1-c3-0-28
Degree 22
Conductor 338338
Sign −1-1
Analytic cond. 19.942619.9426
Root an. cond. 4.465714.46571
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  − 2·2-s + 0.405·3-s + 4·4-s + 6.36·5-s − 0.811·6-s − 2.55·7-s − 8·8-s − 26.8·9-s − 12.7·10-s + 26.1·11-s + 1.62·12-s + 5.10·14-s + 2.58·15-s + 16·16-s − 93.7·17-s + 53.6·18-s − 37.2·19-s + 25.4·20-s − 1.03·21-s − 52.2·22-s − 104.·23-s − 3.24·24-s − 84.5·25-s − 21.8·27-s − 10.2·28-s + 249.·29-s − 5.16·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.0780·3-s + 0.5·4-s + 0.569·5-s − 0.0552·6-s − 0.137·7-s − 0.353·8-s − 0.993·9-s − 0.402·10-s + 0.715·11-s + 0.0390·12-s + 0.0973·14-s + 0.0444·15-s + 0.250·16-s − 1.33·17-s + 0.702·18-s − 0.449·19-s + 0.284·20-s − 0.0107·21-s − 0.506·22-s − 0.951·23-s − 0.0276·24-s − 0.676·25-s − 0.155·27-s − 0.0688·28-s + 1.59·29-s − 0.0314·30-s + ⋯

Functional equation

Λ(s)=(338s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(338s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 338338    =    2⋅1322 \cdot 13^{2}
Sign: −1-1
Analytic conductor: 19.942619.9426
Root analytic conductor: 4.465714.46571
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 338, ( :3/2), −1)(2,\ 338,\ (\ :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+2T 1 + 2T
13 1 1
good3 1−0.405T+27T2 1 - 0.405T + 27T^{2}
5 1−6.36T+125T2 1 - 6.36T + 125T^{2}
7 1+2.55T+343T2 1 + 2.55T + 343T^{2}
11 1−26.1T+1.33e3T2 1 - 26.1T + 1.33e3T^{2}
17 1+93.7T+4.91e3T2 1 + 93.7T + 4.91e3T^{2}
19 1+37.2T+6.85e3T2 1 + 37.2T + 6.85e3T^{2}
23 1+104.T+1.21e4T2 1 + 104.T + 1.21e4T^{2}
29 1−249.T+2.43e4T2 1 - 249.T + 2.43e4T^{2}
31 1−278.T+2.97e4T2 1 - 278.T + 2.97e4T^{2}
37 1+10.9T+5.06e4T2 1 + 10.9T + 5.06e4T^{2}
41 1+371.T+6.89e4T2 1 + 371.T + 6.89e4T^{2}
43 1+413.T+7.95e4T2 1 + 413.T + 7.95e4T^{2}
47 1−238.T+1.03e5T2 1 - 238.T + 1.03e5T^{2}
53 1+424.T+1.48e5T2 1 + 424.T + 1.48e5T^{2}
59 1+774.T+2.05e5T2 1 + 774.T + 2.05e5T^{2}
61 1+123.T+2.26e5T2 1 + 123.T + 2.26e5T^{2}
67 1+881.T+3.00e5T2 1 + 881.T + 3.00e5T^{2}
71 1−118.T+3.57e5T2 1 - 118.T + 3.57e5T^{2}
73 1−209.T+3.89e5T2 1 - 209.T + 3.89e5T^{2}
79 1+532.T+4.93e5T2 1 + 532.T + 4.93e5T^{2}
83 1+376.T+5.71e5T2 1 + 376.T + 5.71e5T^{2}
89 1−42.6T+7.04e5T2 1 - 42.6T + 7.04e5T^{2}
97 1−639.T+9.12e5T2 1 - 639.T + 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

−10.48507400452114034455344179041, −9.688010731060488302240715097642, −8.742054684412510920450851833998, −8.139381257562754021363752272394, −6.58389709471778843864988520433, −6.14345367119653406161522408975, −4.59548535936550969534478016969, −2.99076026862635851518010660339, −1.78465533015960456163356550174, 0, 1.78465533015960456163356550174, 2.99076026862635851518010660339, 4.59548535936550969534478016969, 6.14345367119653406161522408975, 6.58389709471778843864988520433, 8.139381257562754021363752272394, 8.742054684412510920450851833998, 9.688010731060488302240715097642, 10.48507400452114034455344179041

Graph of the ZZ-function along the critical line