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Properties

Label 2-338-1.1-c3-0-35
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 − 3.66·3-s + 4·4-s + 8.53·5-s − 7.32·6-s − 4.20·7-s + 8·8-s − 13.5·9-s + 17.0·10-s − 65.3·11-s − 14.6·12-s − 8.40·14-s − 31.2·15-s + 16·16-s − 26.9·17-s − 27.1·18-s − 13.3·19-s + 34.1·20-s + 15.3·21-s − 130.·22-s + 159.·23-s − 29.3·24-s − 52.2·25-s + 148.·27-s − 16.8·28-s − 301.·29-s − 62.5·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.705·3-s + 0.5·4-s + 0.763·5-s − 0.498·6-s − 0.226·7-s + 0.353·8-s − 0.502·9-s + 0.539·10-s − 1.79·11-s − 0.352·12-s − 0.160·14-s − 0.538·15-s + 0.250·16-s − 0.383·17-s − 0.355·18-s − 0.161·19-s + 0.381·20-s + 0.159·21-s − 1.26·22-s + 1.44·23-s − 0.249·24-s − 0.417·25-s + 1.05·27-s − 0.113·28-s − 1.92·29-s − 0.380·30-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+3.66T+27T2 1 + 3.66T + 27T^{2}
5 1−8.53T+125T2 1 - 8.53T + 125T^{2}
7 1+4.20T+343T2 1 + 4.20T + 343T^{2}
11 1+65.3T+1.33e3T2 1 + 65.3T + 1.33e3T^{2}
17 1+26.9T+4.91e3T2 1 + 26.9T + 4.91e3T^{2}
19 1+13.3T+6.85e3T2 1 + 13.3T + 6.85e3T^{2}
23 1−159.T+1.21e4T2 1 - 159.T + 1.21e4T^{2}
29 1+301.T+2.43e4T2 1 + 301.T + 2.43e4T^{2}
31 1+73.0T+2.97e4T2 1 + 73.0T + 2.97e4T^{2}
37 1+118.T+5.06e4T2 1 + 118.T + 5.06e4T^{2}
41 1+432.T+6.89e4T2 1 + 432.T + 6.89e4T^{2}
43 1+356.T+7.95e4T2 1 + 356.T + 7.95e4T^{2}
47 1−588.T+1.03e5T2 1 - 588.T + 1.03e5T^{2}
53 1+269.T+1.48e5T2 1 + 269.T + 1.48e5T^{2}
59 1−230.T+2.05e5T2 1 - 230.T + 2.05e5T^{2}
61 1+380.T+2.26e5T2 1 + 380.T + 2.26e5T^{2}
67 1−435.T+3.00e5T2 1 - 435.T + 3.00e5T^{2}
71 1−65.9T+3.57e5T2 1 - 65.9T + 3.57e5T^{2}
73 1−885.T+3.89e5T2 1 - 885.T + 3.89e5T^{2}
79 1+385.T+4.93e5T2 1 + 385.T + 4.93e5T^{2}
83 1+254.T+5.71e5T2 1 + 254.T + 5.71e5T^{2}
89 1+372.T+7.04e5T2 1 + 372.T + 7.04e5T^{2}
97 1−1.31e3T+9.12e5T2 1 - 1.31e3T + 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

−10.83379361544133899419068836958, −10.03740675992481756951953953006, −8.806940937433261807111198860763, −7.54340686659322814899718423354, −6.46492298251966230174502980093, −5.45325304350041968606725447628, −5.07258063890915693944331603274, −3.27645644032136196654995203915, −2.10461573118887613239567151674, 0, 2.10461573118887613239567151674, 3.27645644032136196654995203915, 5.07258063890915693944331603274, 5.45325304350041968606725447628, 6.46492298251966230174502980093, 7.54340686659322814899718423354, 8.806940937433261807111198860763, 10.03740675992481756951953953006, 10.83379361544133899419068836958

Graph of the ZZ-function along the critical line