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Properties

Label 2-60e2-4.3-c0-0-1
Degree 22
Conductor 36003600
Sign 0.5−0.866i0.5 - 0.866i
Analytic cond. 1.796631.79663
Root an. cond. 1.340381.34038
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  + 1.73i·7-s + 13-s − 1.73i·19-s + 1.73i·31-s + 2·37-s + 1.73i·43-s − 1.99·49-s − 61-s + 1.73i·67-s − 2·73-s + 1.73i·91-s + 97-s + 109-s + ⋯
L(s)  = 1  + 1.73i·7-s + 13-s − 1.73i·19-s + 1.73i·31-s + 2·37-s + 1.73i·43-s − 1.99·49-s − 61-s + 1.73i·67-s − 2·73-s + 1.73i·91-s + 97-s + 109-s + ⋯

Functional equation

Λ(s)=(3600s/2 ΓC(s) L(s)=((0.5−0.866i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3600s/2 ΓC(s) L(s)=((0.5−0.866i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 36003600    =    24⋅32⋅522^{4} \cdot 3^{2} \cdot 5^{2}
Sign: 0.5−0.866i0.5 - 0.866i
Analytic conductor: 1.796631.79663
Root analytic conductor: 1.340381.34038
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3600(3151,⋅)\chi_{3600} (3151, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3600, ( :0), 0.5−0.866i)(2,\ 3600,\ (\ :0),\ 0.5 - 0.866i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3045417341.304541734
L(12)L(\frac12) ≈\approx 1.3045417341.304541734
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 1
3 1 1
5 1 1
good7 1−1.73iT−T2 1 - 1.73iT - T^{2}
11 1−T2 1 - T^{2}
13 1−T+T2 1 - T + T^{2}
17 1+T2 1 + T^{2}
19 1+1.73iT−T2 1 + 1.73iT - T^{2}
23 1−T2 1 - T^{2}
29 1+T2 1 + T^{2}
31 1−1.73iT−T2 1 - 1.73iT - T^{2}
37 1−2T+T2 1 - 2T + T^{2}
41 1+T2 1 + T^{2}
43 1−1.73iT−T2 1 - 1.73iT - T^{2}
47 1−T2 1 - T^{2}
53 1+T2 1 + T^{2}
59 1−T2 1 - T^{2}
61 1+T+T2 1 + T + T^{2}
67 1−1.73iT−T2 1 - 1.73iT - T^{2}
71 1−T2 1 - T^{2}
73 1+2T+T2 1 + 2T + T^{2}
79 1−T2 1 - T^{2}
83 1−T2 1 - T^{2}
89 1+T2 1 + T^{2}
97 1−T+T2 1 - T + 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

−8.867470330904173342821232056490, −8.352937788875759211196280498756, −7.39502693616187491454632631477, −6.41932832409070900917269557885, −5.96532382474648780660299390712, −5.10156698455657904980448646174, −4.41151213172758678057851222304, −3.06362229102243986605066833457, −2.61659039085290545781143792099, −1.36858935655896603797884666969, 0.849287628200568591356291987110, 1.88976216955315927353710890275, 3.35147616920509415702730398561, 3.96765265705382171201653111116, 4.53293702398748791021384047300, 5.85717804278547754481236016725, 6.25529837166572875014060175006, 7.34431156178424980269602863901, 7.74150735439921101569748939819, 8.446156141950173316095183383653

Graph of the ZZ-function along the critical line