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Properties

Label 2-45e2-5.4-c1-0-25
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  + 2·4-s + 2i·7-s + 3·11-s + 4i·13-s + 4·16-s + 6i·17-s + 19-s − 6i·23-s + 4i·28-s − 9·29-s − 31-s + 8i·37-s − 3·41-s + 4i·43-s + 6·44-s + ⋯
L(s)  = 1  + 4-s + 0.755i·7-s + 0.904·11-s + 1.10i·13-s + 16-s + 1.45i·17-s + 0.229·19-s − 1.25i·23-s + 0.755i·28-s − 1.67·29-s − 0.179·31-s + 1.31i·37-s − 0.468·41-s + 0.609i·43-s + 0.904·44-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

\(L(1)\) \(\approx\) \(2.301858413\)
\(L(\frac12)\) \(\approx\) \(2.301858413\)
\(L(\frac{3}{2})\) 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 \( 1 - 2T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 12iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 3T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 14iT - 67T^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 + 4iT - 97T^{2} \)
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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

−9.141453528492942704595709107973, −8.595917921580745693787108849374, −7.68348048555598805857180133237, −6.70656156337577338450575889374, −6.31755624275359754380337576290, −5.49245375669920536201161866018, −4.26901777903278307552380391079, −3.42504147038113013085880278452, −2.22212161856329540190229865614, −1.54642921574642177238387606234, 0.816034237419382064912545105533, 1.98091267472640435563170042041, 3.19944623222774795090184095586, 3.80456597956703903027291671532, 5.15313189926702788425457926705, 5.83998847462150188180702483096, 6.81770418813623450899480633344, 7.46594192810939365406692402198, 7.85459564367845933724048728252, 9.273707276885932272852470470505

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