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Properties

Label 2-45e2-5.4-c1-0-51
Degree $2$
Conductor $2025$
Sign $0.894 + 0.447i$
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  + 1.47i·2-s − 0.170·4-s − 3.86i·7-s + 2.69i·8-s − 0.260·11-s − 4.07i·13-s + 5.69·14-s − 4.31·16-s + 3.26i·17-s − 4.24·19-s − 0.383i·22-s − 8.69i·23-s + 6.00·26-s + 0.659i·28-s + 4.22·29-s + ⋯
L(s)  = 1  + 1.04i·2-s − 0.0852·4-s − 1.46i·7-s + 0.952i·8-s − 0.0784·11-s − 1.13i·13-s + 1.52·14-s − 1.07·16-s + 0.790i·17-s − 0.974·19-s − 0.0817i·22-s − 1.81i·23-s + 1.17·26-s + 0.124i·28-s + 0.784·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
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: \((2,\ 2025,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.489085978\)
\(L(\frac12)\) \(\approx\) \(1.489085978\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.47iT - 2T^{2} \)
7 \( 1 + 3.86iT - 7T^{2} \)
11 \( 1 + 0.260T + 11T^{2} \)
13 \( 1 + 4.07iT - 13T^{2} \)
17 \( 1 - 3.26iT - 17T^{2} \)
19 \( 1 + 4.24T + 19T^{2} \)
23 \( 1 + 8.69iT - 23T^{2} \)
29 \( 1 - 4.22T + 29T^{2} \)
31 \( 1 - 2.65T + 31T^{2} \)
37 \( 1 - 2.27iT - 37T^{2} \)
41 \( 1 + 5.64T + 41T^{2} \)
43 \( 1 + 9.07iT - 43T^{2} \)
47 \( 1 + 1.42iT - 47T^{2} \)
53 \( 1 + 11.3iT - 53T^{2} \)
59 \( 1 + 7.12T + 59T^{2} \)
61 \( 1 - 2.52T + 61T^{2} \)
67 \( 1 + 11.2iT - 67T^{2} \)
71 \( 1 - 8.38T + 71T^{2} \)
73 \( 1 - 0.403iT - 73T^{2} \)
79 \( 1 - 3.04T + 79T^{2} \)
83 \( 1 + 4.58iT - 83T^{2} \)
89 \( 1 - 7.17T + 89T^{2} \)
97 \( 1 + 3.11iT - 97T^{2} \)
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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

−8.524620245884783697155677108226, −8.287520491783871752949913020769, −7.46540424446044080354643204627, −6.64066051882586472158026326106, −6.28033020242165309313715625724, −5.11852963313212241760211981541, −4.40922086790597980928944870180, −3.35914670082575413046274042638, −2.11031957597352366927071960112, −0.51340984455350772406430294947, 1.44667075724397311053468892669, 2.34180808569065060823998864637, 3.02418969554951450189862058689, 4.12129563407493342557994222490, 5.05263376916019583327976772806, 6.07253045611321709557931432095, 6.72808000742378059858935429929, 7.71358721682718649514315261818, 8.775636015971072467068480567224, 9.328181488363872552515250291508

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