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Properties

Label 2-1368-1368.43-c0-0-1
Degree 22
Conductor 13681368
Sign −0.378−0.925i-0.378 - 0.925i
Analytic cond. 0.6827200.682720
Root an. cond. 0.8262690.826269
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.5 − 0.866i)3-s + (−0.939 − 0.342i)4-s + (−0.939 + 0.342i)6-s + (−0.5 + 0.866i)8-s + (−0.499 + 0.866i)9-s + (−0.766 − 1.32i)11-s + (0.173 + 0.984i)12-s + (0.766 + 0.642i)16-s + (−1.87 + 0.684i)17-s + (0.766 + 0.642i)18-s + (−0.939 − 0.342i)19-s + (−1.43 + 0.524i)22-s + 0.999·24-s + (−0.939 − 0.342i)25-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.5 − 0.866i)3-s + (−0.939 − 0.342i)4-s + (−0.939 + 0.342i)6-s + (−0.5 + 0.866i)8-s + (−0.499 + 0.866i)9-s + (−0.766 − 1.32i)11-s + (0.173 + 0.984i)12-s + (0.766 + 0.642i)16-s + (−1.87 + 0.684i)17-s + (0.766 + 0.642i)18-s + (−0.939 − 0.342i)19-s + (−1.43 + 0.524i)22-s + 0.999·24-s + (−0.939 − 0.342i)25-s + ⋯

Functional equation

Λ(s)=(1368s/2 ΓC(s) L(s)=((−0.378−0.925i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1368s/2 ΓC(s) L(s)=((−0.378−0.925i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13681368    =    23⋅32⋅192^{3} \cdot 3^{2} \cdot 19
Sign: −0.378−0.925i-0.378 - 0.925i
Analytic conductor: 0.6827200.682720
Root analytic conductor: 0.8262690.826269
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1368(43,⋅)\chi_{1368} (43, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1368, ( :0), −0.378−0.925i)(2,\ 1368,\ (\ :0),\ -0.378 - 0.925i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.34437080330.3443708033
L(12)L(\frac12) ≈\approx 0.34437080330.3443708033
L(1)L(1) 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+(−0.173+0.984i)T 1 + (-0.173 + 0.984i)T
3 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
19 1+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
good5 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
7 1−T2 1 - T^{2}
11 1+(0.766+1.32i)T+(−0.5+0.866i)T2 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2}
13 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
17 1+(1.87−0.684i)T+(0.766−0.642i)T2 1 + (1.87 - 0.684i)T + (0.766 - 0.642i)T^{2}
23 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
29 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1−T2 1 - T^{2}
41 1+(0.326−0.118i)T+(0.766−0.642i)T2 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2}
43 1+(−0.939+0.342i)T+(0.766−0.642i)T2 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2}
47 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
53 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
59 1+(1.43+1.20i)T+(0.173+0.984i)T2 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2}
61 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
67 1+(−0.0603−0.342i)T+(−0.939+0.342i)T2 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2}
71 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
73 1+(−0.266+1.50i)T+(−0.939−0.342i)T2 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2}
79 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
83 1−1.53T+T2 1 - 1.53T + T^{2}
89 1+(0.173+0.984i)T+(−0.939+0.342i)T2 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2}
97 1+(0.326−1.85i)T+(−0.939−0.342i)T2 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{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

−9.148925545010146224376032503324, −8.497733848829218322058313484326, −7.83899819123040920448136564942, −6.48180455460170379161220832893, −5.93907368015460238500615358177, −4.96588727201958137153427373380, −3.99275793384514625725584890439, −2.69635582228939759072176510442, −1.91448329070623219941270031189, −0.27112283062176131331796015235, 2.47993740918818229724802701289, 4.00667315732449319582489333184, 4.53483647896850311120643275300, 5.28874848635526302501875890500, 6.21693689657049994559068136518, 6.96591821606840412908047919435, 7.80133745228839099078454474672, 8.852446703593935934162251072908, 9.390259849240522267763564603313, 10.19884929831162974546686500935

Graph of the ZZ-function along the critical line