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Properties

Label 2-45e2-5.4-c1-0-49
Degree 22
Conductor 20252025
Sign 0.447+0.894i0.447 + 0.894i
Analytic cond. 16.169716.1697
Root an. cond. 4.021154.02115
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 2·4-s − 5·11-s + 4i·13-s − 4·16-s − 4i·17-s + 5·19-s − 10i·22-s − 6i·23-s − 8·26-s − 5·29-s − 9·31-s − 8i·32-s + 8·34-s + 10i·37-s + 10i·38-s + ⋯
L(s)  = 1  + 1.41i·2-s − 4-s − 1.50·11-s + 1.10i·13-s − 16-s − 0.970i·17-s + 1.14·19-s − 2.13i·22-s − 1.25i·23-s − 1.56·26-s − 0.928·29-s − 1.61·31-s − 1.41i·32-s + 1.37·34-s + 1.64i·37-s + 1.62i·38-s + ⋯

Functional equation

Λ(s)=(2025s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2025s/2 ΓC(s+1/2) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 20252025    =    34⋅523^{4} \cdot 5^{2}
Sign: 0.447+0.894i0.447 + 0.894i
Analytic conductor: 16.169716.1697
Root analytic conductor: 4.021154.02115
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2025(649,⋅)\chi_{2025} (649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 2025, ( :1/2), 0.447+0.894i)(2,\ 2025,\ (\ :1/2),\ 0.447 + 0.894i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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)
bad3 1 1
5 1 1
good2 1−2iT−2T2 1 - 2iT - 2T^{2}
7 1−7T2 1 - 7T^{2}
11 1+5T+11T2 1 + 5T + 11T^{2}
13 1−4iT−13T2 1 - 4iT - 13T^{2}
17 1+4iT−17T2 1 + 4iT - 17T^{2}
19 1−5T+19T2 1 - 5T + 19T^{2}
23 1+6iT−23T2 1 + 6iT - 23T^{2}
29 1+5T+29T2 1 + 5T + 29T^{2}
31 1+9T+31T2 1 + 9T + 31T^{2}
37 1−10iT−37T2 1 - 10iT - 37T^{2}
41 1+7T+41T2 1 + 7T + 41T^{2}
43 1+2iT−43T2 1 + 2iT - 43T^{2}
47 1−2iT−47T2 1 - 2iT - 47T^{2}
53 1+8iT−53T2 1 + 8iT - 53T^{2}
59 1+T+59T2 1 + T + 59T^{2}
61 1+2T+61T2 1 + 2T + 61T^{2}
67 1+6iT−67T2 1 + 6iT - 67T^{2}
71 1+T+71T2 1 + T + 71T^{2}
73 1+8iT−73T2 1 + 8iT - 73T^{2}
79 1+12T+79T2 1 + 12T + 79T^{2}
83 1+6iT−83T2 1 + 6iT - 83T^{2}
89 1+9T+89T2 1 + 9T + 89T^{2}
97 1+14iT−97T2 1 + 14iT - 97T^{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

−8.796757103659955882184750512393, −8.033181550631331755586115812966, −7.28821866874789272140674721459, −6.87971926453899959377091503690, −5.83493122272084627541816387112, −5.15552066072193444170338386412, −4.56791650727092598518936736394, −3.16139029288812680023044000618, −2.05394936162397305553317567306, 0, 1.41367908081764886833082515348, 2.45010986367889804375991163493, 3.30246899933004115351570197787, 3.95899605993864123804780182089, 5.38310096248894106468536261852, 5.61615316638313121438803924235, 7.26881987381792152724521784217, 7.70116598363739981728345139898, 8.745638184639293078654162317345

Graph of the ZZ-function along the critical line