Properties

Label 2-1425-5.4-c1-0-8
Degree $2$
Conductor $1425$
Sign $-0.894 + 0.447i$
Analytic cond. $11.3786$
Root an. cond. $3.37323$
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  + 2i·2-s i·3-s − 2·4-s + 2·6-s + 5i·7-s − 9-s + 11-s + 2i·12-s + 2i·13-s − 10·14-s − 4·16-s + i·17-s − 2i·18-s + 19-s + 5·21-s + 2i·22-s + ⋯
L(s)  = 1  + 1.41i·2-s − 0.577i·3-s − 4-s + 0.816·6-s + 1.88i·7-s − 0.333·9-s + 0.301·11-s + 0.577i·12-s + 0.554i·13-s − 2.67·14-s − 16-s + 0.242i·17-s − 0.471i·18-s + 0.229·19-s + 1.09·21-s + 0.426i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1425\)    =    \(3 \cdot 5^{2} \cdot 19\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(11.3786\)
Root analytic conductor: \(3.37323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1425} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1425,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.213258106\)
\(L(\frac12)\) \(\approx\) \(1.213258106\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
19 \( 1 - T \)
good2 \( 1 - 2iT - 2T^{2} \)
7 \( 1 - 5iT - 7T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - iT - 17T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 - 9iT - 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 11iT - 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−9.413536435927328250175960390265, −8.919360813747291069389790560515, −8.333437702861934967332757652589, −7.51918595249373066857123824590, −6.62360351443807592976756319500, −6.03442508510505748600502576477, −5.44590430097588137327209655755, −4.49112833241380922224768471874, −2.85061582980062512808896909837, −1.89387420170770766800380050930, 0.48458776575377232738719977766, 1.59333984936815195897336748818, 3.07564879122154319068832060598, 3.75895657612691884878267342316, 4.36002682787085093088138193913, 5.41081426384085138124804158955, 6.82464734474922463008327271495, 7.43488676905309528588313224795, 8.559541758158458095367011329787, 9.548766757501507698246424256944

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