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Properties

Label 2-338-1.1-c7-0-7
Degree 22
Conductor 338338
Sign 11
Analytic cond. 105.586105.586
Root an. cond. 10.275510.2755
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s − 39.9·3-s + 64·4-s − 323.·5-s + 319.·6-s + 568.·7-s − 512·8-s − 588.·9-s + 2.58e3·10-s − 238.·11-s − 2.55e3·12-s − 4.54e3·14-s + 1.29e4·15-s + 4.09e3·16-s + 2.04e4·17-s + 4.70e3·18-s − 9.64e3·19-s − 2.07e4·20-s − 2.27e4·21-s + 1.90e3·22-s − 7.82e4·23-s + 2.04e4·24-s + 2.66e4·25-s + 1.10e5·27-s + 3.63e4·28-s − 1.38e5·29-s − 1.03e5·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.855·3-s + 0.5·4-s − 1.15·5-s + 0.604·6-s + 0.626·7-s − 0.353·8-s − 0.268·9-s + 0.818·10-s − 0.0539·11-s − 0.427·12-s − 0.443·14-s + 0.990·15-s + 0.250·16-s + 1.01·17-s + 0.190·18-s − 0.322·19-s − 0.579·20-s − 0.535·21-s + 0.0381·22-s − 1.34·23-s + 0.302·24-s + 0.341·25-s + 1.08·27-s + 0.313·28-s − 1.05·29-s − 0.700·30-s + ⋯

Functional equation

Λ(s)=(338s/2 ΓC(s) L(s)=( Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(338s/2 ΓC(s+7/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 338338    =    2⋅1322 \cdot 13^{2}
Sign: 11
Analytic conductor: 105.586105.586
Root analytic conductor: 10.275510.2755
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 338, ( :7/2), 1)(2,\ 338,\ (\ :7/2),\ 1)

Particular Values

L(4)L(4) ≈\approx 0.34925091640.3492509164
L(12)L(\frac12) ≈\approx 0.34925091640.3492509164
L(92)L(\frac{9}{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+8T 1 + 8T
13 1 1
good3 1+39.9T+2.18e3T2 1 + 39.9T + 2.18e3T^{2}
5 1+323.T+7.81e4T2 1 + 323.T + 7.81e4T^{2}
7 1−568.T+8.23e5T2 1 - 568.T + 8.23e5T^{2}
11 1+238.T+1.94e7T2 1 + 238.T + 1.94e7T^{2}
17 1−2.04e4T+4.10e8T2 1 - 2.04e4T + 4.10e8T^{2}
19 1+9.64e3T+8.93e8T2 1 + 9.64e3T + 8.93e8T^{2}
23 1+7.82e4T+3.40e9T2 1 + 7.82e4T + 3.40e9T^{2}
29 1+1.38e5T+1.72e10T2 1 + 1.38e5T + 1.72e10T^{2}
31 1−1.60e5T+2.75e10T2 1 - 1.60e5T + 2.75e10T^{2}
37 1+1.52e5T+9.49e10T2 1 + 1.52e5T + 9.49e10T^{2}
41 1−1.85e5T+1.94e11T2 1 - 1.85e5T + 1.94e11T^{2}
43 1−8.50e4T+2.71e11T2 1 - 8.50e4T + 2.71e11T^{2}
47 1+1.20e6T+5.06e11T2 1 + 1.20e6T + 5.06e11T^{2}
53 1+6.65e5T+1.17e12T2 1 + 6.65e5T + 1.17e12T^{2}
59 1+2.48e6T+2.48e12T2 1 + 2.48e6T + 2.48e12T^{2}
61 1+3.04e6T+3.14e12T2 1 + 3.04e6T + 3.14e12T^{2}
67 1+3.87e5T+6.06e12T2 1 + 3.87e5T + 6.06e12T^{2}
71 1−3.68e6T+9.09e12T2 1 - 3.68e6T + 9.09e12T^{2}
73 1−1.57e6T+1.10e13T2 1 - 1.57e6T + 1.10e13T^{2}
79 1−2.29e6T+1.92e13T2 1 - 2.29e6T + 1.92e13T^{2}
83 1+7.93e6T+2.71e13T2 1 + 7.93e6T + 2.71e13T^{2}
89 1−8.15e6T+4.42e13T2 1 - 8.15e6T + 4.42e13T^{2}
97 1+1.33e6T+8.07e13T2 1 + 1.33e6T + 8.07e13T^{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

−10.55872724301638611268295221969, −9.456033705737520298846435036877, −8.055269230927260252564143723358, −7.926980555370721384110610346743, −6.56363104845749508210945193692, −5.57867484772998308280544509253, −4.44324204928085912764662509741, −3.20527214781971589569875194324, −1.60443714002249210641926510294, −0.33240100619318930520040079901, 0.33240100619318930520040079901, 1.60443714002249210641926510294, 3.20527214781971589569875194324, 4.44324204928085912764662509741, 5.57867484772998308280544509253, 6.56363104845749508210945193692, 7.926980555370721384110610346743, 8.055269230927260252564143723358, 9.456033705737520298846435036877, 10.55872724301638611268295221969

Graph of the ZZ-function along the critical line