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Properties

Label 2-338-1.1-c3-0-10
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 − 8.86·3-s + 4·4-s − 16.8·5-s + 17.7·6-s − 10.8·7-s − 8·8-s + 51.5·9-s + 33.7·10-s + 35.1·11-s − 35.4·12-s + 21.7·14-s + 149.·15-s + 16·16-s + 30.3·17-s − 103.·18-s + 28.3·19-s − 67.4·20-s + 96.3·21-s − 70.3·22-s − 24.6·23-s + 70.9·24-s + 159.·25-s − 218.·27-s − 43.4·28-s + 290.·29-s − 299.·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.70·3-s + 0.5·4-s − 1.50·5-s + 1.20·6-s − 0.586·7-s − 0.353·8-s + 1.91·9-s + 1.06·10-s + 0.964·11-s − 0.853·12-s + 0.414·14-s + 2.57·15-s + 0.250·16-s + 0.432·17-s − 1.35·18-s + 0.342·19-s − 0.754·20-s + 1.00·21-s − 0.682·22-s − 0.223·23-s + 0.603·24-s + 1.27·25-s − 1.55·27-s − 0.293·28-s + 1.85·29-s − 1.81·30-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+8.86T+27T2 1 + 8.86T + 27T^{2}
5 1+16.8T+125T2 1 + 16.8T + 125T^{2}
7 1+10.8T+343T2 1 + 10.8T + 343T^{2}
11 1−35.1T+1.33e3T2 1 - 35.1T + 1.33e3T^{2}
17 1−30.3T+4.91e3T2 1 - 30.3T + 4.91e3T^{2}
19 1−28.3T+6.85e3T2 1 - 28.3T + 6.85e3T^{2}
23 1+24.6T+1.21e4T2 1 + 24.6T + 1.21e4T^{2}
29 1−290.T+2.43e4T2 1 - 290.T + 2.43e4T^{2}
31 1+219.T+2.97e4T2 1 + 219.T + 2.97e4T^{2}
37 1+118.T+5.06e4T2 1 + 118.T + 5.06e4T^{2}
41 1+83.6T+6.89e4T2 1 + 83.6T + 6.89e4T^{2}
43 1−293.T+7.95e4T2 1 - 293.T + 7.95e4T^{2}
47 1−166.T+1.03e5T2 1 - 166.T + 1.03e5T^{2}
53 1+76.3T+1.48e5T2 1 + 76.3T + 1.48e5T^{2}
59 1+184.T+2.05e5T2 1 + 184.T + 2.05e5T^{2}
61 1−197.T+2.26e5T2 1 - 197.T + 2.26e5T^{2}
67 1+321.T+3.00e5T2 1 + 321.T + 3.00e5T^{2}
71 1−368.T+3.57e5T2 1 - 368.T + 3.57e5T^{2}
73 1+843.T+3.89e5T2 1 + 843.T + 3.89e5T^{2}
79 1−184.T+4.93e5T2 1 - 184.T + 4.93e5T^{2}
83 1+1.27e3T+5.71e5T2 1 + 1.27e3T + 5.71e5T^{2}
89 1−1.36e3T+7.04e5T2 1 - 1.36e3T + 7.04e5T^{2}
97 1−690.T+9.12e5T2 1 - 690.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

−10.80548420381294106932270366918, −9.972954005536728734943577911733, −8.802698797584223237786149020416, −7.56287049042641503345827577619, −6.83800568967202682896615138911, −5.96486402895414887560376149598, −4.63285351153799202281635447035, −3.52344255058554531854459970568, −1.03152159367182028220753386786, 0, 1.03152159367182028220753386786, 3.52344255058554531854459970568, 4.63285351153799202281635447035, 5.96486402895414887560376149598, 6.83800568967202682896615138911, 7.56287049042641503345827577619, 8.802698797584223237786149020416, 9.972954005536728734943577911733, 10.80548420381294106932270366918

Graph of the ZZ-function along the critical line