Properties

Label 2-45e2-5.4-c1-0-53
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s + 4-s − 3i·7-s − 3i·8-s + 2·11-s + 2i·13-s − 3·14-s − 16-s − 4i·17-s + 8·19-s − 2i·22-s + 3i·23-s + 2·26-s − 3i·28-s − 29-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.5·4-s − 1.13i·7-s − 1.06i·8-s + 0.603·11-s + 0.554i·13-s − 0.801·14-s − 0.250·16-s − 0.970i·17-s + 1.83·19-s − 0.426i·22-s + 0.625i·23-s + 0.392·26-s − 0.566i·28-s − 0.185·29-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: \(0\)
Selberg data: \((2,\ 2025,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.226823184\)
\(L(\frac12)\) \(\approx\) \(2.226823184\)
\(L(\frac{3}{2})\) 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 \( 1 + iT - 2T^{2} \)
7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 - 3iT - 23T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 + 7iT - 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 + 14T + 59T^{2} \)
61 \( 1 - 7T + 61T^{2} \)
67 \( 1 + 3iT - 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−9.376741619708044686081009119557, −7.943172169087852326465739355537, −7.15597677106333918247873720315, −6.85542743206508358566982511708, −5.69687811853695285147992883523, −4.61755931869771490478403951750, −3.68037413688321536704585762883, −3.05726363888766807250643789256, −1.72838282775858378684817411189, −0.835991572129583043879403521465, 1.48514295959288926795682074441, 2.60315331284799429850266084139, 3.46407843211430183907768555112, 4.87293775156618201827285958311, 5.65579691115611211232008148623, 6.14943180023679515630897611102, 7.02473703030165987105606274057, 7.80973990263537273930878371406, 8.504759777416897739586786736708, 9.163013812254466075787784776182

Graph of the $Z$-function along the critical line