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Properties

Label 2-45e2-5.4-c1-0-29
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  + 0.732i·2-s + 1.46·4-s + 4.73i·7-s + 2.53i·8-s + 5.73·11-s + 1.46i·13-s − 3.46·14-s + 1.07·16-s − 2.73i·17-s − 4.46·19-s + 4.19i·22-s + 3.46i·23-s − 1.07·26-s + 6.92i·28-s + 3.19·29-s + ⋯
L(s)  = 1  + 0.517i·2-s + 0.732·4-s + 1.78i·7-s + 0.896i·8-s + 1.72·11-s + 0.406i·13-s − 0.925·14-s + 0.267·16-s − 0.662i·17-s − 1.02·19-s + 0.894i·22-s + 0.722i·23-s − 0.210·26-s + 1.30i·28-s + 0.593·29-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

\(L(1)\) \(\approx\) \(2.329221163\)
\(L(\frac12)\) \(\approx\) \(2.329221163\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.732iT - 2T^{2} \)
7 \( 1 - 4.73iT - 7T^{2} \)
11 \( 1 - 5.73T + 11T^{2} \)
13 \( 1 - 1.46iT - 13T^{2} \)
17 \( 1 + 2.73iT - 17T^{2} \)
19 \( 1 + 4.46T + 19T^{2} \)
23 \( 1 - 3.46iT - 23T^{2} \)
29 \( 1 - 3.19T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 - 2.73iT - 37T^{2} \)
41 \( 1 - 7.19T + 41T^{2} \)
43 \( 1 - 0.196iT - 43T^{2} \)
47 \( 1 + 8.73iT - 47T^{2} \)
53 \( 1 + 6.73iT - 53T^{2} \)
59 \( 1 + 8.26T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 - 3.73T + 71T^{2} \)
73 \( 1 + 7.66iT - 73T^{2} \)
79 \( 1 + 15.4T + 79T^{2} \)
83 \( 1 + 2.19iT - 83T^{2} \)
89 \( 1 - 5.19T + 89T^{2} \)
97 \( 1 - 9.66iT - 97T^{2} \)
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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

−9.020616105888920011428479968997, −8.812086578354417372638567553362, −7.81845949848757794078266046776, −6.80399041775547669686780702238, −6.32550839166849040564627089699, −5.66119752295333949580766848680, −4.75152011131746763193572668082, −3.50497279141247677389639022210, −2.42905742205240507649360004244, −1.67071166874190612974472950840, 0.848246867776011159903908519155, 1.65930912530023178915196283739, 3.00273124164700206240772816749, 4.14369124628295645538307192909, 4.17678166205214135320890635040, 6.02265698693532791772044017157, 6.60413775222749163698229206804, 7.20905065089437835313981681639, 7.999796119894216333146818077731, 9.026744784726519087497594617186

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