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Properties

Label 2-1800-200.37-c0-0-1
Degree 22
Conductor 18001800
Sign 0.827+0.562i0.827 + 0.562i
Analytic cond. 0.8983170.898317
Root an. cond. 0.9477950.947795
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  + (0.987 − 0.156i)2-s + (0.951 − 0.309i)4-s + (0.156 + 0.987i)5-s + (−1.26 − 1.26i)7-s + (0.891 − 0.453i)8-s + (0.309 + 0.951i)10-s + (1.16 − 1.59i)11-s + (−1.44 − 1.04i)14-s + (0.809 − 0.587i)16-s + (0.453 + 0.891i)20-s + (0.896 − 1.76i)22-s + (−0.951 + 0.309i)25-s + (−1.58 − 0.809i)28-s + (0.437 + 1.34i)29-s + (−0.587 + 1.80i)31-s + (0.707 − 0.707i)32-s + ⋯
L(s)  = 1  + (0.987 − 0.156i)2-s + (0.951 − 0.309i)4-s + (0.156 + 0.987i)5-s + (−1.26 − 1.26i)7-s + (0.891 − 0.453i)8-s + (0.309 + 0.951i)10-s + (1.16 − 1.59i)11-s + (−1.44 − 1.04i)14-s + (0.809 − 0.587i)16-s + (0.453 + 0.891i)20-s + (0.896 − 1.76i)22-s + (−0.951 + 0.309i)25-s + (−1.58 − 0.809i)28-s + (0.437 + 1.34i)29-s + (−0.587 + 1.80i)31-s + (0.707 − 0.707i)32-s + ⋯

Functional equation

Λ(s)=(1800s/2 ΓC(s) L(s)=((0.827+0.562i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1800s/2 ΓC(s) L(s)=((0.827+0.562i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18001800    =    23⋅32⋅522^{3} \cdot 3^{2} \cdot 5^{2}
Sign: 0.827+0.562i0.827 + 0.562i
Analytic conductor: 0.8983170.898317
Root analytic conductor: 0.9477950.947795
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1800(37,⋅)\chi_{1800} (37, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1800, ( :0), 0.827+0.562i)(2,\ 1800,\ (\ :0),\ 0.827 + 0.562i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.0407224372.040722437
L(12)L(\frac12) ≈\approx 2.0407224372.040722437
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+(−0.987+0.156i)T 1 + (-0.987 + 0.156i)T
3 1 1
5 1+(−0.156−0.987i)T 1 + (-0.156 - 0.987i)T
good7 1+(1.26+1.26i)T+iT2 1 + (1.26 + 1.26i)T + iT^{2}
11 1+(−1.16+1.59i)T+(−0.309−0.951i)T2 1 + (-1.16 + 1.59i)T + (-0.309 - 0.951i)T^{2}
13 1+(0.951+0.309i)T2 1 + (0.951 + 0.309i)T^{2}
17 1+(0.587−0.809i)T2 1 + (0.587 - 0.809i)T^{2}
19 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
23 1+(0.951−0.309i)T2 1 + (0.951 - 0.309i)T^{2}
29 1+(−0.437−1.34i)T+(−0.809+0.587i)T2 1 + (-0.437 - 1.34i)T + (-0.809 + 0.587i)T^{2}
31 1+(0.587−1.80i)T+(−0.809−0.587i)T2 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2}
37 1+(−0.951−0.309i)T2 1 + (-0.951 - 0.309i)T^{2}
41 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
43 1+iT2 1 + iT^{2}
47 1+(−0.587−0.809i)T2 1 + (-0.587 - 0.809i)T^{2}
53 1+(−0.533+1.04i)T+(−0.587−0.809i)T2 1 + (-0.533 + 1.04i)T + (-0.587 - 0.809i)T^{2}
59 1+(0.253−0.183i)T+(0.309−0.951i)T2 1 + (0.253 - 0.183i)T + (0.309 - 0.951i)T^{2}
61 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
67 1+(−0.587+0.809i)T2 1 + (-0.587 + 0.809i)T^{2}
71 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
73 1+(1.39−0.221i)T+(0.951−0.309i)T2 1 + (1.39 - 0.221i)T + (0.951 - 0.309i)T^{2}
79 1+(1.11−0.363i)T+(0.809−0.587i)T2 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2}
83 1+(−0.550+0.280i)T+(0.587−0.809i)T2 1 + (-0.550 + 0.280i)T + (0.587 - 0.809i)T^{2}
89 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
97 1+(0.412−0.809i)T+(−0.587−0.809i)T2 1 + (0.412 - 0.809i)T + (-0.587 - 0.809i)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

−9.638935967367129521031434268299, −8.643645633122293659701677026578, −7.31722375379722224009977606962, −6.75905058679485614292429353346, −6.35797071278338639551206506530, −5.46336435155787424586311140702, −4.08757700973510664690000723265, −3.39606869110169052523420321160, −3.03939895569984911033418252825, −1.28256989055274864656887971436, 1.82503051765450891098729033790, 2.65005575802921222235232569941, 3.96892197884780457403399458638, 4.51095668271484310525002847950, 5.63082471685750181578129078185, 6.10601852937052284052218404371, 6.91331008421149839833974544952, 7.83752928366725268925100048692, 8.897227820879674167235534799861, 9.529644692907562302252056924808

Graph of the ZZ-function along the critical line