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Properties

Label 2-45e2-5.4-c1-0-49
Degree $2$
Conductor $2025$
Sign $0.447 + 0.894i$
Analytic cond. $16.1697$
Root an. cond. $4.02115$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

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

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

Invariants

Degree: \(2\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(16.1697\)
Root analytic conductor: \(4.02115\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2025} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2025,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 - 2iT - 2T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 5T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + 9T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + 7T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 + 8iT - 53T^{2} \)
59 \( 1 + T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 6iT - 67T^{2} \)
71 \( 1 + T + 71T^{2} \)
73 \( 1 + 8iT - 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line