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Properties

Label 2-1425-5.4-c1-0-24
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 + 3.64i·7-s + 7.93i·8-s − 9-s + 5.64·11-s + 5.00i·12-s − 5.64i·13-s + 9.64·14-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 + 1.37i·7-s + 2.80i·8-s − 0.333·9-s + 1.70·11-s + 1.44i·12-s − 1.56i·13-s + 2.57·14-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

\(L(1)\) \(\approx\) \(1.515545968\)
\(L(\frac12)\) \(\approx\) \(1.515545968\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.64iT - 7T^{2} \)
11 \( 1 - 5.64T + 11T^{2} \)
13 \( 1 + 5.64iT - 13T^{2} \)
17 \( 1 - 4iT - 17T^{2} \)
23 \( 1 + 1.29iT - 23T^{2} \)
29 \( 1 - 6.93T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 1.64iT - 37T^{2} \)
41 \( 1 + 4.35T + 41T^{2} \)
43 \( 1 - 0.354iT - 43T^{2} \)
47 \( 1 + 9.29iT - 47T^{2} \)
53 \( 1 - 0.708iT - 53T^{2} \)
59 \( 1 + 0.708T + 59T^{2} \)
61 \( 1 + 0.708T + 61T^{2} \)
67 \( 1 + 14.5iT - 67T^{2} \)
71 \( 1 + 3.29T + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 - 1.06T + 89T^{2} \)
97 \( 1 + 12.9iT - 97T^{2} \)
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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

−9.360143914069802781754932282640, −8.495845679306433854419339447038, −8.256913737782010434068908369797, −6.54170038930594026168364181781, −5.70881982029232643084603607904, −4.76076014315921208722936644579, −3.58859458748282625006603076806, −2.85540761259727915283466745800, −1.89395016087929708957627954000, −0.884996695984924135403672373460, 1.00592871596478144731837563201, 3.56865178314112894591400262752, 4.42609621640050870761065964335, 4.67173632818568316080806737483, 6.08465483102187199736904327314, 6.77306864104276983433946453203, 7.14085292957193922158697181859, 8.153003260342570139826533489236, 9.084262970852396108513543404233, 9.450271011683333976194610807093

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