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Properties

Label 2-1800-5.4-c3-0-8
Degree 22
Conductor 18001800
Sign −0.447−0.894i-0.447 - 0.894i
Analytic cond. 106.203106.203
Root an. cond. 10.305510.3055
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  + 5i·7-s − 14·11-s − i·13-s − 46i·17-s − 19·19-s − 46i·23-s + 14·29-s + 133·31-s + 258i·37-s − 84·41-s + 167i·43-s − 410i·47-s + 318·49-s + 456i·53-s − 194·59-s + ⋯
L(s)  = 1  + 0.269i·7-s − 0.383·11-s − 0.0213i·13-s − 0.656i·17-s − 0.229·19-s − 0.417i·23-s + 0.0896·29-s + 0.770·31-s + 1.14i·37-s − 0.319·41-s + 0.592i·43-s − 1.27i·47-s + 0.927·49-s + 1.18i·53-s − 0.428·59-s + ⋯

Functional equation

Λ(s)=(1800s/2 ΓC(s) L(s)=((−0.447−0.894i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1800s/2 ΓC(s+3/2) L(s)=((−0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18001800    =    23⋅32⋅522^{3} \cdot 3^{2} \cdot 5^{2}
Sign: −0.447−0.894i-0.447 - 0.894i
Analytic conductor: 106.203106.203
Root analytic conductor: 10.305510.3055
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1800(649,⋅)\chi_{1800} (649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1800, ( :3/2), −0.447−0.894i)(2,\ 1800,\ (\ :3/2),\ -0.447 - 0.894i)

Particular Values

L(2)L(2) ≈\approx 1.0068661871.006866187
L(12)L(\frac12) ≈\approx 1.0068661871.006866187
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 1
5 1 1
good7 1−5iT−343T2 1 - 5iT - 343T^{2}
11 1+14T+1.33e3T2 1 + 14T + 1.33e3T^{2}
13 1+iT−2.19e3T2 1 + iT - 2.19e3T^{2}
17 1+46iT−4.91e3T2 1 + 46iT - 4.91e3T^{2}
19 1+19T+6.85e3T2 1 + 19T + 6.85e3T^{2}
23 1+46iT−1.21e4T2 1 + 46iT - 1.21e4T^{2}
29 1−14T+2.43e4T2 1 - 14T + 2.43e4T^{2}
31 1−133T+2.97e4T2 1 - 133T + 2.97e4T^{2}
37 1−258iT−5.06e4T2 1 - 258iT - 5.06e4T^{2}
41 1+84T+6.89e4T2 1 + 84T + 6.89e4T^{2}
43 1−167iT−7.95e4T2 1 - 167iT - 7.95e4T^{2}
47 1+410iT−1.03e5T2 1 + 410iT - 1.03e5T^{2}
53 1−456iT−1.48e5T2 1 - 456iT - 1.48e5T^{2}
59 1+194T+2.05e5T2 1 + 194T + 2.05e5T^{2}
61 1+17T+2.26e5T2 1 + 17T + 2.26e5T^{2}
67 1−653iT−3.00e5T2 1 - 653iT - 3.00e5T^{2}
71 1+828T+3.57e5T2 1 + 828T + 3.57e5T^{2}
73 1+570iT−3.89e5T2 1 + 570iT - 3.89e5T^{2}
79 1−552T+4.93e5T2 1 - 552T + 4.93e5T^{2}
83 1−142iT−5.71e5T2 1 - 142iT - 5.71e5T^{2}
89 1+1.10e3T+7.04e5T2 1 + 1.10e3T + 7.04e5T^{2}
97 1−841iT−9.12e5T2 1 - 841iT - 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

−9.082133900033455591715144742662, −8.433450826693909415053468609062, −7.62286755274128728592367266603, −6.78660211770914979109623115383, −5.97713656241324522250851016807, −5.07628384929594705147211156638, −4.32212075797690691869831929344, −3.12784551227140321830914791744, −2.34057615370752309759200190601, −1.03859485927879428294446752631, 0.23136645742809323516223790514, 1.50150977452343665718071761984, 2.59410291728196429053821993872, 3.65745635718767971466762032472, 4.48464602346167380237537251035, 5.45846272535351158481225122956, 6.23860887190096125143912679361, 7.13212757852434674979840553204, 7.88061089006616393460637474878, 8.617190674675728636225261170691

Graph of the ZZ-function along the critical line