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Properties

Label 2-1425-5.4-c1-0-0
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  + 2.41i·2-s + i·3-s − 3.82·4-s − 2.41·6-s + 3.41i·7-s − 4.41i·8-s − 9-s − 1.41·11-s − 3.82i·12-s − 2.58i·13-s − 8.24·14-s + 2.99·16-s − 6.82i·17-s − 2.41i·18-s − 19-s + ⋯
L(s)  = 1  + 1.70i·2-s + 0.577i·3-s − 1.91·4-s − 0.985·6-s + 1.29i·7-s − 1.56i·8-s − 0.333·9-s − 0.426·11-s − 1.10i·12-s − 0.717i·13-s − 2.20·14-s + 0.749·16-s − 1.65i·17-s − 0.569i·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

\(L(1)\) \(\approx\) \(0.2224758159\)
\(L(\frac12)\) \(\approx\) \(0.2224758159\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.41iT - 2T^{2} \)
7 \( 1 - 3.41iT - 7T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 + 2.58iT - 13T^{2} \)
17 \( 1 + 6.82iT - 17T^{2} \)
23 \( 1 - 3.65iT - 23T^{2} \)
29 \( 1 + 5.07T + 29T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 + 3.07iT - 37T^{2} \)
41 \( 1 + 4.58T + 41T^{2} \)
43 \( 1 + 3.41iT - 43T^{2} \)
47 \( 1 - 11.6iT - 47T^{2} \)
53 \( 1 + 4iT - 53T^{2} \)
59 \( 1 - 8.48T + 59T^{2} \)
61 \( 1 + 5.65T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + 6.48iT - 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
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

−9.777479712105243101426554903868, −9.190487232450484959335053419400, −8.679603612569919888614368352683, −7.72024848632860527460778424818, −7.17080134243982280545657683482, −5.99755040317724892408994811045, −5.37408684470790528660445431016, −5.03338798173030763925376351624, −3.68797103428698016705989957426, −2.48398024978209773095816678685, 0.089029927205885183509843580473, 1.45148502431081731912984660102, 2.17208860997407972119271264366, 3.57875389035724105245981503095, 4.00909195946539138799715523001, 5.10304137887260366736252639084, 6.39455478338696639706121626170, 7.26695776196807122973908873750, 8.215143043763422286961935567596, 8.982161756735445040775025004401

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