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Properties

Label 2-338-1.1-c7-0-29
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 − 71.1·3-s + 64·4-s + 523.·5-s + 569.·6-s − 416.·7-s − 512·8-s + 2.87e3·9-s − 4.18e3·10-s + 3.42e3·11-s − 4.55e3·12-s + 3.33e3·14-s − 3.72e4·15-s + 4.09e3·16-s − 6.53e3·17-s − 2.30e4·18-s + 2.79e4·19-s + 3.35e4·20-s + 2.96e4·21-s − 2.73e4·22-s + 1.06e5·23-s + 3.64e4·24-s + 1.95e5·25-s − 4.90e4·27-s − 2.66e4·28-s + 6.85e4·29-s + 2.98e5·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.52·3-s + 0.5·4-s + 1.87·5-s + 1.07·6-s − 0.459·7-s − 0.353·8-s + 1.31·9-s − 1.32·10-s + 0.775·11-s − 0.760·12-s + 0.324·14-s − 2.84·15-s + 0.250·16-s − 0.322·17-s − 0.930·18-s + 0.935·19-s + 0.936·20-s + 0.698·21-s − 0.548·22-s + 1.81·23-s + 0.537·24-s + 2.50·25-s − 0.479·27-s − 0.229·28-s + 0.521·29-s + 2.01·30-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 1.4997471731.499747173
L(12)L(\frac12) ≈\approx 1.4997471731.499747173
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+71.1T+2.18e3T2 1 + 71.1T + 2.18e3T^{2}
5 1−523.T+7.81e4T2 1 - 523.T + 7.81e4T^{2}
7 1+416.T+8.23e5T2 1 + 416.T + 8.23e5T^{2}
11 1−3.42e3T+1.94e7T2 1 - 3.42e3T + 1.94e7T^{2}
17 1+6.53e3T+4.10e8T2 1 + 6.53e3T + 4.10e8T^{2}
19 1−2.79e4T+8.93e8T2 1 - 2.79e4T + 8.93e8T^{2}
23 1−1.06e5T+3.40e9T2 1 - 1.06e5T + 3.40e9T^{2}
29 1−6.85e4T+1.72e10T2 1 - 6.85e4T + 1.72e10T^{2}
31 1−5.15e4T+2.75e10T2 1 - 5.15e4T + 2.75e10T^{2}
37 1−4.81e4T+9.49e10T2 1 - 4.81e4T + 9.49e10T^{2}
41 1+6.02e5T+1.94e11T2 1 + 6.02e5T + 1.94e11T^{2}
43 1−9.16e5T+2.71e11T2 1 - 9.16e5T + 2.71e11T^{2}
47 1+3.26e5T+5.06e11T2 1 + 3.26e5T + 5.06e11T^{2}
53 1−9.34e5T+1.17e12T2 1 - 9.34e5T + 1.17e12T^{2}
59 1−1.17e6T+2.48e12T2 1 - 1.17e6T + 2.48e12T^{2}
61 1+2.89e6T+3.14e12T2 1 + 2.89e6T + 3.14e12T^{2}
67 1−3.18e5T+6.06e12T2 1 - 3.18e5T + 6.06e12T^{2}
71 1−1.28e6T+9.09e12T2 1 - 1.28e6T + 9.09e12T^{2}
73 1+1.67e6T+1.10e13T2 1 + 1.67e6T + 1.10e13T^{2}
79 1+8.09e6T+1.92e13T2 1 + 8.09e6T + 1.92e13T^{2}
83 1−5.57e6T+2.71e13T2 1 - 5.57e6T + 2.71e13T^{2}
89 1−7.87e6T+4.42e13T2 1 - 7.87e6T + 4.42e13T^{2}
97 1+6.80e6T+8.07e13T2 1 + 6.80e6T + 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

−10.31712903758051365512273099383, −9.563515161715840445579318589154, −8.913614366650806546256328320824, −7.01339613694661659856620335418, −6.46821681891209597350460649470, −5.70975673505344896551169212456, −4.88923701035266010361412830962, −2.87267500947223221718814603193, −1.47826896211398055483076364582, −0.77963620483203718354731703557, 0.77963620483203718354731703557, 1.47826896211398055483076364582, 2.87267500947223221718814603193, 4.88923701035266010361412830962, 5.70975673505344896551169212456, 6.46821681891209597350460649470, 7.01339613694661659856620335418, 8.913614366650806546256328320824, 9.563515161715840445579318589154, 10.31712903758051365512273099383

Graph of the ZZ-function along the critical line