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Properties

Label 2-45e2-5.4-c1-0-19
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  + 1.63i·2-s − 0.669·4-s + 0.505i·7-s + 2.17i·8-s + 3.10·11-s + 6.23i·13-s − 0.825·14-s − 4.89·16-s − 6.10i·17-s + 5.57·19-s + 5.06i·22-s + 3.82i·23-s − 10.1·26-s − 0.338i·28-s − 2.45·29-s + ⋯
L(s)  = 1  + 1.15i·2-s − 0.334·4-s + 0.191i·7-s + 0.768i·8-s + 0.934·11-s + 1.73i·13-s − 0.220·14-s − 1.22·16-s − 1.47i·17-s + 1.27·19-s + 1.07i·22-s + 0.797i·23-s − 1.99·26-s − 0.0638i·28-s − 0.456·29-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

\(L(1)\) \(\approx\) \(1.920688350\)
\(L(\frac12)\) \(\approx\) \(1.920688350\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.63iT - 2T^{2} \)
7 \( 1 - 0.505iT - 7T^{2} \)
11 \( 1 - 3.10T + 11T^{2} \)
13 \( 1 - 6.23iT - 13T^{2} \)
17 \( 1 + 6.10iT - 17T^{2} \)
19 \( 1 - 5.57T + 19T^{2} \)
23 \( 1 - 3.82iT - 23T^{2} \)
29 \( 1 + 2.45T + 29T^{2} \)
31 \( 1 - 4.22T + 31T^{2} \)
37 \( 1 - 6.72iT - 37T^{2} \)
41 \( 1 + 5.44T + 41T^{2} \)
43 \( 1 + 1.32iT - 43T^{2} \)
47 \( 1 + 3.70iT - 47T^{2} \)
53 \( 1 + 2.54iT - 53T^{2} \)
59 \( 1 + 2.88T + 59T^{2} \)
61 \( 1 + 2.84T + 61T^{2} \)
67 \( 1 - 2.40iT - 67T^{2} \)
71 \( 1 - 5.54T + 71T^{2} \)
73 \( 1 - 11.7iT - 73T^{2} \)
79 \( 1 + 3.40T + 79T^{2} \)
83 \( 1 - 13.9iT - 83T^{2} \)
89 \( 1 + 3.38T + 89T^{2} \)
97 \( 1 - 11.0iT - 97T^{2} \)
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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

−9.313845758537882002822121641334, −8.658029531172382700184778999345, −7.70637712566014949040932551075, −6.88778168837358551127769524564, −6.68630380065311129013695989227, −5.55347396633613289782871439938, −4.92140266173881322183223189813, −3.92880529227357721177855879555, −2.68505374313997770400563664893, −1.46261698546784323059281181563, 0.72575795276903801257747343921, 1.69143183919361226886144494268, 2.91769773678406536251495639913, 3.56803492592950249179983138441, 4.40330579365661897029433689559, 5.62189493455816455901493116862, 6.36375576129382308338160679636, 7.33999217938790386824635381114, 8.122397974260851623690740922285, 9.034348626760276112955625819774

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