Properties

Label 2-1425-5.4-c1-0-14
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  + 1.73i·2-s + i·3-s − 0.999·4-s − 1.73·6-s + 2.73i·7-s + 1.73i·8-s − 9-s + 4.73·11-s − 0.999i·12-s + 0.732i·13-s − 4.73·14-s − 5·16-s − 1.73i·18-s − 19-s − 2.73·21-s + 8.19i·22-s + ⋯
L(s)  = 1  + 1.22i·2-s + 0.577i·3-s − 0.499·4-s − 0.707·6-s + 1.03i·7-s + 0.612i·8-s − 0.333·9-s + 1.42·11-s − 0.288i·12-s + 0.203i·13-s − 1.26·14-s − 1.25·16-s − 0.408i·18-s − 0.229·19-s − 0.596·21-s + 1.74i·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

\(L(1)\) \(\approx\) \(1.681800650\)
\(L(\frac12)\) \(\approx\) \(1.681800650\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT - 2T^{2} \)
7 \( 1 - 2.73iT - 7T^{2} \)
11 \( 1 - 4.73T + 11T^{2} \)
13 \( 1 - 0.732iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
23 \( 1 - 3.46iT - 23T^{2} \)
29 \( 1 + 8.19T + 29T^{2} \)
31 \( 1 - 8.92T + 31T^{2} \)
37 \( 1 - 6.19iT - 37T^{2} \)
41 \( 1 - 1.26T + 41T^{2} \)
43 \( 1 - 4.19iT - 43T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 + 9.46iT - 53T^{2} \)
59 \( 1 + 2.53T + 59T^{2} \)
61 \( 1 + 6.53T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 4.39T + 71T^{2} \)
73 \( 1 + 16.9iT - 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + 12.9iT - 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 - 6.19iT - 97T^{2} \)
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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

−9.504540385799054678589774400983, −9.166012906438243983710981132370, −8.369352885101763285156165379661, −7.59189307104467560316044389735, −6.47261347679879916694530541751, −6.12182128120132405028418973598, −5.18134402907165270604921705342, −4.36962830412626668665455935424, −3.18637560587312674456491414749, −1.86742139228939138811610787981, 0.69572251448223840059976131504, 1.58816059526215800369514723902, 2.70614984713408326198989218159, 3.84110742579424141279933260404, 4.31325486800042197658098992426, 5.87886160778175022649779287555, 6.81468346204564592302521781382, 7.28413680150958422239649983214, 8.446243127903146220558604137285, 9.306653885206434229879097741082

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