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Properties

Label 2-338-1.1-c3-0-1
Degree 22
Conductor 338338
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3.92·3-s + 4·4-s + 0.152·5-s + 7.85·6-s − 33.9·7-s − 8·8-s − 11.5·9-s − 0.305·10-s + 10.5·11-s − 15.7·12-s + 67.9·14-s − 0.600·15-s + 16·16-s − 41.3·17-s + 23.1·18-s − 131.·19-s + 0.611·20-s + 133.·21-s − 21.0·22-s + 161.·23-s + 31.4·24-s − 124.·25-s + 151.·27-s − 135.·28-s − 35.6·29-s + 1.20·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.755·3-s + 0.5·4-s + 0.0136·5-s + 0.534·6-s − 1.83·7-s − 0.353·8-s − 0.428·9-s − 0.00966·10-s + 0.288·11-s − 0.377·12-s + 1.29·14-s − 0.0103·15-s + 0.250·16-s − 0.589·17-s + 0.303·18-s − 1.58·19-s + 0.00683·20-s + 1.38·21-s − 0.204·22-s + 1.46·23-s + 0.267·24-s − 0.999·25-s + 1.07·27-s − 0.917·28-s − 0.227·29-s + 0.00730·30-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: 11
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: 00
Selberg data: (2, 338, ( :3/2), 1)(2,\ 338,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.38695803410.3869580341
L(12)L(\frac12) ≈\approx 0.38695803410.3869580341
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+3.92T+27T2 1 + 3.92T + 27T^{2}
5 1−0.152T+125T2 1 - 0.152T + 125T^{2}
7 1+33.9T+343T2 1 + 33.9T + 343T^{2}
11 1−10.5T+1.33e3T2 1 - 10.5T + 1.33e3T^{2}
17 1+41.3T+4.91e3T2 1 + 41.3T + 4.91e3T^{2}
19 1+131.T+6.85e3T2 1 + 131.T + 6.85e3T^{2}
23 1−161.T+1.21e4T2 1 - 161.T + 1.21e4T^{2}
29 1+35.6T+2.43e4T2 1 + 35.6T + 2.43e4T^{2}
31 1+12.7T+2.97e4T2 1 + 12.7T + 2.97e4T^{2}
37 1+183.T+5.06e4T2 1 + 183.T + 5.06e4T^{2}
41 1+443.T+6.89e4T2 1 + 443.T + 6.89e4T^{2}
43 1−466.T+7.95e4T2 1 - 466.T + 7.95e4T^{2}
47 1−282.T+1.03e5T2 1 - 282.T + 1.03e5T^{2}
53 1−114.T+1.48e5T2 1 - 114.T + 1.48e5T^{2}
59 1−703.T+2.05e5T2 1 - 703.T + 2.05e5T^{2}
61 1−600.T+2.26e5T2 1 - 600.T + 2.26e5T^{2}
67 1−542.T+3.00e5T2 1 - 542.T + 3.00e5T^{2}
71 1+907.T+3.57e5T2 1 + 907.T + 3.57e5T^{2}
73 1−498.T+3.89e5T2 1 - 498.T + 3.89e5T^{2}
79 1+356.T+4.93e5T2 1 + 356.T + 4.93e5T^{2}
83 1−934.T+5.71e5T2 1 - 934.T + 5.71e5T^{2}
89 1−581.T+7.04e5T2 1 - 581.T + 7.04e5T^{2}
97 1+334.T+9.12e5T2 1 + 334.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

−10.92962701016954645093673092311, −10.25578724325014370807910245186, −9.244435875527658408003966361508, −8.587604786426500736412597466496, −6.95708181118247005341886598230, −6.47777420604391620635191641724, −5.52709357529250059901980916223, −3.80783396679750249252387123125, −2.50553495336628906387022987414, −0.45547467819414588104636695648, 0.45547467819414588104636695648, 2.50553495336628906387022987414, 3.80783396679750249252387123125, 5.52709357529250059901980916223, 6.47777420604391620635191641724, 6.95708181118247005341886598230, 8.587604786426500736412597466496, 9.244435875527658408003966361508, 10.25578724325014370807910245186, 10.92962701016954645093673092311

Graph of the ZZ-function along the critical line