Properties

Label 2-1425-5.4-c1-0-23
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  + 0.414i·2-s i·3-s + 1.82·4-s + 0.414·6-s − 0.585i·7-s + 1.58i·8-s − 9-s + 1.41·11-s − 1.82i·12-s + 5.41i·13-s + 0.242·14-s + 3·16-s + 1.17i·17-s − 0.414i·18-s − 19-s + ⋯
L(s)  = 1  + 0.292i·2-s − 0.577i·3-s + 0.914·4-s + 0.169·6-s − 0.221i·7-s + 0.560i·8-s − 0.333·9-s + 0.426·11-s − 0.527i·12-s + 1.50i·13-s + 0.0648·14-s + 0.750·16-s + 0.284i·17-s − 0.0976i·18-s − 0.229·19-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

\(L(1)\) \(\approx\) \(2.181157687\)
\(L(\frac12)\) \(\approx\) \(2.181157687\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.414iT - 2T^{2} \)
7 \( 1 + 0.585iT - 7T^{2} \)
11 \( 1 - 1.41T + 11T^{2} \)
13 \( 1 - 5.41iT - 13T^{2} \)
17 \( 1 - 1.17iT - 17T^{2} \)
23 \( 1 - 7.65iT - 23T^{2} \)
29 \( 1 - 9.07T + 29T^{2} \)
31 \( 1 - 6.48T + 31T^{2} \)
37 \( 1 + 11.0iT - 37T^{2} \)
41 \( 1 + 7.41T + 41T^{2} \)
43 \( 1 - 0.585iT - 43T^{2} \)
47 \( 1 + 0.343iT - 47T^{2} \)
53 \( 1 - 4iT - 53T^{2} \)
59 \( 1 + 8.48T + 59T^{2} \)
61 \( 1 - 5.65T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + 4.48T + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 + 10.4iT - 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 - 4.24iT - 97T^{2} \)
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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

−9.506208788781433868785124921646, −8.668802540539766980978641603542, −7.80500771801853056830614327480, −7.04553013939272355791961732358, −6.51426015492510743831309948680, −5.77783309763764189384947224967, −4.57452053714312857723160308504, −3.45341836792722375098256444944, −2.24750099196501803270251030260, −1.36762343570010205407982450013, 0.966363272200579114527837998645, 2.59785400670633621853230051223, 3.10818226997182733522973607819, 4.35048729250825573292645590290, 5.28158357826624201873487015586, 6.30895510446541431267484849454, 6.83450472382660994723731136000, 8.157980922106515212143734569210, 8.491211610833132304317202629377, 9.853749845624629233316471477466

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