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Properties

Label 2-1425-5.4-c1-0-4
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  + 2.64i·2-s − i·3-s − 5.00·4-s + 2.64·6-s − 1.64i·7-s − 7.93i·8-s − 9-s + 0.354·11-s + 5.00i·12-s − 0.354i·13-s + 4.35·14-s + 11.0·16-s + 4i·17-s − 2.64i·18-s + 19-s + ⋯
L(s)  = 1  + 1.87i·2-s − 0.577i·3-s − 2.50·4-s + 1.08·6-s − 0.622i·7-s − 2.80i·8-s − 0.333·9-s + 0.106·11-s + 1.44i·12-s − 0.0982i·13-s + 1.16·14-s + 2.75·16-s + 0.970i·17-s − 0.623i·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

\(L(1)\) \(\approx\) \(0.7535482456\)
\(L(\frac12)\) \(\approx\) \(0.7535482456\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.64iT - 2T^{2} \)
7 \( 1 + 1.64iT - 7T^{2} \)
11 \( 1 - 0.354T + 11T^{2} \)
13 \( 1 + 0.354iT - 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
23 \( 1 - 9.29iT - 23T^{2} \)
29 \( 1 + 8.93T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + 3.64iT - 37T^{2} \)
41 \( 1 + 9.64T + 41T^{2} \)
43 \( 1 - 5.64iT - 43T^{2} \)
47 \( 1 - 1.29iT - 47T^{2} \)
53 \( 1 - 11.2iT - 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 - 6.58iT - 67T^{2} \)
71 \( 1 - 7.29T + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + 6.58T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 - 16.9T + 89T^{2} \)
97 \( 1 - 2.93iT - 97T^{2} \)
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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

−9.566197283392287097447848998737, −8.954773302630430381247419669096, −7.920562302100694830952547002371, −7.60552874571891845520799414961, −6.84659128051662348087991197991, −6.00884736493140656197035006263, −5.43909063193821190254668619828, −4.33575912099777727312945573373, −3.46530521443920727581844780315, −1.35917482622367926127006304528, 0.32673721334488739356012600522, 1.93039957994342878797821849001, 2.81803512599027198934523148644, 3.61954067693656378523645027394, 4.66642961651913353231932962902, 5.17008533026142241298646497908, 6.42035845323397024182214404453, 7.909254499103621116219769804310, 8.892487044417369655234754916948, 9.180381506936532248878288031303

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