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Properties

Label 2-1344-8.5-c3-0-14
Degree 22
Conductor 13441344
Sign −0.707−0.707i-0.707 - 0.707i
Analytic cond. 79.298579.2985
Root an. cond. 8.904978.90497
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s + 11.7i·5-s + 7·7-s − 9·9-s + 72.4i·11-s + 50.7i·13-s + 35.2·15-s + 60.4·17-s + 33.8i·19-s − 21i·21-s + 116.·23-s − 13.1·25-s + 27i·27-s + 13.1i·29-s − 250.·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 1.05i·5-s + 0.377·7-s − 0.333·9-s + 1.98i·11-s + 1.08i·13-s + 0.607·15-s + 0.862·17-s + 0.408i·19-s − 0.218i·21-s + 1.05·23-s − 0.105·25-s + 0.192i·27-s + 0.0844i·29-s − 1.44·31-s + ⋯

Functional equation

Λ(s)=(1344s/2 ΓC(s) L(s)=((−0.707−0.707i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1344s/2 ΓC(s+3/2) L(s)=((−0.707−0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13441344    =    26⋅3⋅72^{6} \cdot 3 \cdot 7
Sign: −0.707−0.707i-0.707 - 0.707i
Analytic conductor: 79.298579.2985
Root analytic conductor: 8.904978.90497
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1344(673,⋅)\chi_{1344} (673, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1344, ( :3/2), −0.707−0.707i)(2,\ 1344,\ (\ :3/2),\ -0.707 - 0.707i)

Particular Values

L(2)L(2) ≈\approx 1.6875372061.687537206
L(12)L(\frac12) ≈\approx 1.6875372061.687537206
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 1
3 1+3iT 1 + 3iT
7 1−7T 1 - 7T
good5 1−11.7iT−125T2 1 - 11.7iT - 125T^{2}
11 1−72.4iT−1.33e3T2 1 - 72.4iT - 1.33e3T^{2}
13 1−50.7iT−2.19e3T2 1 - 50.7iT - 2.19e3T^{2}
17 1−60.4T+4.91e3T2 1 - 60.4T + 4.91e3T^{2}
19 1−33.8iT−6.85e3T2 1 - 33.8iT - 6.85e3T^{2}
23 1−116.T+1.21e4T2 1 - 116.T + 1.21e4T^{2}
29 1−13.1iT−2.43e4T2 1 - 13.1iT - 2.43e4T^{2}
31 1+250.T+2.97e4T2 1 + 250.T + 2.97e4T^{2}
37 1+92.7iT−5.06e4T2 1 + 92.7iT - 5.06e4T^{2}
41 1+69.1T+6.89e4T2 1 + 69.1T + 6.89e4T^{2}
43 1−69.6iT−7.95e4T2 1 - 69.6iT - 7.95e4T^{2}
47 1+346.T+1.03e5T2 1 + 346.T + 1.03e5T^{2}
53 1+585.iT−1.48e5T2 1 + 585. iT - 1.48e5T^{2}
59 1−66.1iT−2.05e5T2 1 - 66.1iT - 2.05e5T^{2}
61 1−492.iT−2.26e5T2 1 - 492. iT - 2.26e5T^{2}
67 1−543.iT−3.00e5T2 1 - 543. iT - 3.00e5T^{2}
71 1+365.T+3.57e5T2 1 + 365.T + 3.57e5T^{2}
73 1−374.T+3.89e5T2 1 - 374.T + 3.89e5T^{2}
79 1+670.T+4.93e5T2 1 + 670.T + 4.93e5T^{2}
83 1+595.iT−5.71e5T2 1 + 595. iT - 5.71e5T^{2}
89 1−1.03e3T+7.04e5T2 1 - 1.03e3T + 7.04e5T^{2}
97 1+218.T+9.12e5T2 1 + 218.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

−9.639653033916567174488846561233, −8.734616664332026526785982283982, −7.54005221899838013245685373643, −7.17328298725753535607104001720, −6.57235464867312573158381614944, −5.39252812304442911460188923447, −4.46465949250789413520861282403, −3.35969749035689206787001635257, −2.21038186443268589270622727610, −1.51376594263855398858736005683, 0.40860185669227677756603194008, 1.20911046786029990205897541867, 2.98036916461063941588924319760, 3.62502650951338126484028111795, 4.94477461824578381953513203218, 5.36734108497429977835857882822, 6.16719014263078314467245894623, 7.58335556545201370265986916267, 8.362599375544047976033886118499, 8.832999577482346417785807433151

Graph of the ZZ-function along the critical line