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Properties

Label 2-338-1.1-c3-0-15
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 − 8.74·3-s + 4·4-s + 14.1·5-s + 17.4·6-s − 28.6·7-s − 8·8-s + 49.4·9-s − 28.3·10-s − 9.49·11-s − 34.9·12-s + 57.3·14-s − 123.·15-s + 16·16-s + 30.6·17-s − 98.8·18-s + 153.·19-s + 56.6·20-s + 250.·21-s + 18.9·22-s − 36.0·23-s + 69.9·24-s + 75.7·25-s − 195.·27-s − 114.·28-s − 49.2·29-s + 247.·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.68·3-s + 0.5·4-s + 1.26·5-s + 1.18·6-s − 1.54·7-s − 0.353·8-s + 1.82·9-s − 0.896·10-s − 0.260·11-s − 0.841·12-s + 1.09·14-s − 2.13·15-s + 0.250·16-s + 0.436·17-s − 1.29·18-s + 1.85·19-s + 0.633·20-s + 2.60·21-s + 0.184·22-s − 0.326·23-s + 0.594·24-s + 0.605·25-s − 1.39·27-s − 0.773·28-s − 0.315·29-s + 1.50·30-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+8.74T+27T2 1 + 8.74T + 27T^{2}
5 1−14.1T+125T2 1 - 14.1T + 125T^{2}
7 1+28.6T+343T2 1 + 28.6T + 343T^{2}
11 1+9.49T+1.33e3T2 1 + 9.49T + 1.33e3T^{2}
17 1−30.6T+4.91e3T2 1 - 30.6T + 4.91e3T^{2}
19 1−153.T+6.85e3T2 1 - 153.T + 6.85e3T^{2}
23 1+36.0T+1.21e4T2 1 + 36.0T + 1.21e4T^{2}
29 1+49.2T+2.43e4T2 1 + 49.2T + 2.43e4T^{2}
31 1−166.T+2.97e4T2 1 - 166.T + 2.97e4T^{2}
37 1+23.8T+5.06e4T2 1 + 23.8T + 5.06e4T^{2}
41 1−125.T+6.89e4T2 1 - 125.T + 6.89e4T^{2}
43 1+434.T+7.95e4T2 1 + 434.T + 7.95e4T^{2}
47 1+186.T+1.03e5T2 1 + 186.T + 1.03e5T^{2}
53 1+400.T+1.48e5T2 1 + 400.T + 1.48e5T^{2}
59 1+408.T+2.05e5T2 1 + 408.T + 2.05e5T^{2}
61 1+603.T+2.26e5T2 1 + 603.T + 2.26e5T^{2}
67 1+287.T+3.00e5T2 1 + 287.T + 3.00e5T^{2}
71 1+961.T+3.57e5T2 1 + 961.T + 3.57e5T^{2}
73 1−963.T+3.89e5T2 1 - 963.T + 3.89e5T^{2}
79 1+1.04e3T+4.93e5T2 1 + 1.04e3T + 4.93e5T^{2}
83 1−313.T+5.71e5T2 1 - 313.T + 5.71e5T^{2}
89 1−675.T+7.04e5T2 1 - 675.T + 7.04e5T^{2}
97 1+272.T+9.12e5T2 1 + 272.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

−10.33486292913899436312384140668, −9.925702154443383169015997146298, −9.314968194200030264335050396064, −7.50834841728014522185643938619, −6.44938967469256768990783514218, −6.00036715608835709021637542600, −5.11453507943084730085066258499, −3.09799543880049557304905635076, −1.31893702792134027444339857185, 0, 1.31893702792134027444339857185, 3.09799543880049557304905635076, 5.11453507943084730085066258499, 6.00036715608835709021637542600, 6.44938967469256768990783514218, 7.50834841728014522185643938619, 9.314968194200030264335050396064, 9.925702154443383169015997146298, 10.33486292913899436312384140668

Graph of the ZZ-function along the critical line