↑

Properties

Label 2-338-13.12-c3-0-18
Degree 22
Conductor 338338
Sign 0.722+0.691i0.722 + 0.691i
Analytic cond. 19.942619.9426
Root an. cond. 4.465714.46571
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s + 0.405·3-s − 4·4-s + 6.36i·5-s − 0.811i·6-s + 2.55i·7-s + 8i·8-s − 26.8·9-s + 12.7·10-s − 26.1i·11-s − 1.62·12-s + 5.10·14-s + 2.58i·15-s + 16·16-s + 93.7·17-s + 53.6i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.0780·3-s − 0.5·4-s + 0.569i·5-s − 0.0552i·6-s + 0.137i·7-s + 0.353i·8-s − 0.993·9-s + 0.402·10-s − 0.715i·11-s − 0.0390·12-s + 0.0973·14-s + 0.0444i·15-s + 0.250·16-s + 1.33·17-s + 0.702i·18-s + ⋯

Functional equation

Λ(s)=(338s/2 ΓC(s) L(s)=((0.722+0.691i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(338s/2 ΓC(s+3/2) L(s)=((0.722+0.691i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 338338    =    2⋅1322 \cdot 13^{2}
Sign: 0.722+0.691i0.722 + 0.691i
Analytic conductor: 19.942619.9426
Root analytic conductor: 4.465714.46571
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ338(337,⋅)\chi_{338} (337, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 338, ( :3/2), 0.722+0.691i)(2,\ 338,\ (\ :3/2),\ 0.722 + 0.691i)

Particular Values

L(2)L(2) ≈\approx 1.6983708251.698370825
L(12)L(\frac12) ≈\approx 1.6983708251.698370825
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+2iT 1 + 2iT
13 1 1
good3 1−0.405T+27T2 1 - 0.405T + 27T^{2}
5 1−6.36iT−125T2 1 - 6.36iT - 125T^{2}
7 1−2.55iT−343T2 1 - 2.55iT - 343T^{2}
11 1+26.1iT−1.33e3T2 1 + 26.1iT - 1.33e3T^{2}
17 1−93.7T+4.91e3T2 1 - 93.7T + 4.91e3T^{2}
19 1+37.2iT−6.85e3T2 1 + 37.2iT - 6.85e3T^{2}
23 1−104.T+1.21e4T2 1 - 104.T + 1.21e4T^{2}
29 1−249.T+2.43e4T2 1 - 249.T + 2.43e4T^{2}
31 1−278.iT−2.97e4T2 1 - 278. iT - 2.97e4T^{2}
37 1−10.9iT−5.06e4T2 1 - 10.9iT - 5.06e4T^{2}
41 1+371.iT−6.89e4T2 1 + 371. iT - 6.89e4T^{2}
43 1−413.T+7.95e4T2 1 - 413.T + 7.95e4T^{2}
47 1+238.iT−1.03e5T2 1 + 238. iT - 1.03e5T^{2}
53 1+424.T+1.48e5T2 1 + 424.T + 1.48e5T^{2}
59 1−774.iT−2.05e5T2 1 - 774. iT - 2.05e5T^{2}
61 1+123.T+2.26e5T2 1 + 123.T + 2.26e5T^{2}
67 1+881.iT−3.00e5T2 1 + 881. iT - 3.00e5T^{2}
71 1−118.iT−3.57e5T2 1 - 118. iT - 3.57e5T^{2}
73 1+209.iT−3.89e5T2 1 + 209. iT - 3.89e5T^{2}
79 1+532.T+4.93e5T2 1 + 532.T + 4.93e5T^{2}
83 1+376.iT−5.71e5T2 1 + 376. iT - 5.71e5T^{2}
89 1+42.6iT−7.04e5T2 1 + 42.6iT - 7.04e5T^{2}
97 1−639.iT−9.12e5T2 1 - 639. iT - 9.12e5T^{2}
show more
show less
   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

−10.86603557112708849793568247720, −10.42778304327317542383022908900, −9.070268192690825095964093058144, −8.486487065459823048382529734529, −7.22281961381026513983673183433, −5.97375981480218879073122648243, −4.98648871879484606777859061549, −3.33872817404783937145454828001, −2.74284704546572871565166159698, −0.873667206228147517130389355106, 0.937018367015572268671032235307, 2.91693750457705846337818534357, 4.39954187606123304998138240213, 5.36771328300417166017375656125, 6.31205144556500458354625524485, 7.56011531358369396022832223982, 8.279651033657509543820840918535, 9.236175173491493525521584389571, 10.05682316083340078986685238240, 11.28091581496642658292213094913

Graph of the ZZ-function along the critical line