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Properties

Label 2-45e2-5.4-c1-0-33
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  + 2.51i·2-s − 4.32·4-s − 0.514i·7-s − 5.83i·8-s − 3.32·11-s + 1.32i·13-s + 1.29·14-s + 6.02·16-s − 3.32i·17-s + 1.32·19-s − 8.34i·22-s − 4.12i·23-s − 3.32·26-s + 2.22i·28-s + 1.38·29-s + ⋯
L(s)  = 1  + 1.77i·2-s − 2.16·4-s − 0.194i·7-s − 2.06i·8-s − 1.00·11-s + 0.366i·13-s + 0.345·14-s + 1.50·16-s − 0.805i·17-s + 0.303·19-s − 1.78i·22-s − 0.860i·23-s − 0.651·26-s + 0.419i·28-s + 0.257·29-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

\(L(1)\) \(\approx\) \(1.117765583\)
\(L(\frac12)\) \(\approx\) \(1.117765583\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.51iT - 2T^{2} \)
7 \( 1 + 0.514iT - 7T^{2} \)
11 \( 1 + 3.32T + 11T^{2} \)
13 \( 1 - 1.32iT - 13T^{2} \)
17 \( 1 + 3.32iT - 17T^{2} \)
19 \( 1 - 1.32T + 19T^{2} \)
23 \( 1 + 4.12iT - 23T^{2} \)
29 \( 1 - 1.38T + 29T^{2} \)
31 \( 1 - 8.73T + 31T^{2} \)
37 \( 1 - 0.292iT - 37T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 + 10.3iT - 43T^{2} \)
47 \( 1 + 4.86iT - 47T^{2} \)
53 \( 1 - 5.02iT - 53T^{2} \)
59 \( 1 - 5.02T + 59T^{2} \)
61 \( 1 - 7.34T + 61T^{2} \)
67 \( 1 - 9.44iT - 67T^{2} \)
71 \( 1 - 8.99T + 71T^{2} \)
73 \( 1 + 6.05iT - 73T^{2} \)
79 \( 1 - 8.05T + 79T^{2} \)
83 \( 1 + 1.54iT - 83T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + 12.2iT - 97T^{2} \)
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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

−8.811959863262951679618031828492, −8.419715684194374661847938023824, −7.55330879301986311824747490388, −6.96224157346523334389417256704, −6.30430129305265561370711469504, −5.28370658167041189357138952202, −4.87581928143703075786869798056, −3.86235977534906597578398889580, −2.52634416830408335780319716787, −0.50382216656992740859692130633, 0.987588809499596905696024685109, 2.13292980739705608243241512653, 2.97022754223013931473739506383, 3.72397012784543579599264249356, 4.77449953728738061567046157465, 5.41211276114262300235222016156, 6.54925391124580554454702641984, 7.919872178311520528979685761225, 8.351599362547051463550175193048, 9.340827390445463177088116827733

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