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Properties

Label 2-338-1.1-c3-0-9
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 − 8.14·3-s + 4·4-s + 12.6·5-s + 16.2·6-s + 28.7·7-s − 8·8-s + 39.3·9-s − 25.3·10-s + 52.8·11-s − 32.5·12-s − 57.5·14-s − 103.·15-s + 16·16-s + 71.0·17-s − 78.7·18-s − 65.7·19-s + 50.6·20-s − 234.·21-s − 105.·22-s − 44.4·23-s + 65.1·24-s + 35.3·25-s − 100.·27-s + 115.·28-s − 78.4·29-s + 206.·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.56·3-s + 0.5·4-s + 1.13·5-s + 1.10·6-s + 1.55·7-s − 0.353·8-s + 1.45·9-s − 0.800·10-s + 1.44·11-s − 0.784·12-s − 1.09·14-s − 1.77·15-s + 0.250·16-s + 1.01·17-s − 1.03·18-s − 0.794·19-s + 0.566·20-s − 2.43·21-s − 1.02·22-s − 0.402·23-s + 0.554·24-s + 0.283·25-s − 0.719·27-s + 0.776·28-s − 0.502·29-s + 1.25·30-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 1.3009516081.300951608
L(12)L(\frac12) ≈\approx 1.3009516081.300951608
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+8.14T+27T2 1 + 8.14T + 27T^{2}
5 1−12.6T+125T2 1 - 12.6T + 125T^{2}
7 1−28.7T+343T2 1 - 28.7T + 343T^{2}
11 1−52.8T+1.33e3T2 1 - 52.8T + 1.33e3T^{2}
17 1−71.0T+4.91e3T2 1 - 71.0T + 4.91e3T^{2}
19 1+65.7T+6.85e3T2 1 + 65.7T + 6.85e3T^{2}
23 1+44.4T+1.21e4T2 1 + 44.4T + 1.21e4T^{2}
29 1+78.4T+2.43e4T2 1 + 78.4T + 2.43e4T^{2}
31 1−195.T+2.97e4T2 1 - 195.T + 2.97e4T^{2}
37 1+233.T+5.06e4T2 1 + 233.T + 5.06e4T^{2}
41 1+120.T+6.89e4T2 1 + 120.T + 6.89e4T^{2}
43 1−405.T+7.95e4T2 1 - 405.T + 7.95e4T^{2}
47 1−400.T+1.03e5T2 1 - 400.T + 1.03e5T^{2}
53 1−116.T+1.48e5T2 1 - 116.T + 1.48e5T^{2}
59 1+781.T+2.05e5T2 1 + 781.T + 2.05e5T^{2}
61 1−52.5T+2.26e5T2 1 - 52.5T + 2.26e5T^{2}
67 1−805.T+3.00e5T2 1 - 805.T + 3.00e5T^{2}
71 1−401.T+3.57e5T2 1 - 401.T + 3.57e5T^{2}
73 1+323.T+3.89e5T2 1 + 323.T + 3.89e5T^{2}
79 1+794.T+4.93e5T2 1 + 794.T + 4.93e5T^{2}
83 1+444.T+5.71e5T2 1 + 444.T + 5.71e5T^{2}
89 1+79.1T+7.04e5T2 1 + 79.1T + 7.04e5T^{2}
97 1+3.89T+9.12e5T2 1 + 3.89T + 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

−11.03020130807953683191247792347, −10.36455951970622868470413873965, −9.449383065648710418690852591542, −8.365904625949980666499908274045, −7.12068216186550186770945567698, −6.12330025802612361833090414777, −5.50480634987000672953430321126, −4.34243423991075556427195790885, −1.85151054322463600776368006137, −1.03028320616562102177409529056, 1.03028320616562102177409529056, 1.85151054322463600776368006137, 4.34243423991075556427195790885, 5.50480634987000672953430321126, 6.12330025802612361833090414777, 7.12068216186550186770945567698, 8.365904625949980666499908274045, 9.449383065648710418690852591542, 10.36455951970622868470413873965, 11.03020130807953683191247792347

Graph of the ZZ-function along the critical line