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Properties

Label 2-1425-285.113-c0-0-1
Degree $2$
Conductor $1425$
Sign $-0.945 + 0.326i$
Analytic cond. $0.711167$
Root an. cond. $0.843307$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s − 1.00·6-s + (−1.22 + 1.22i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s − 1.73i·14-s + 1.00·16-s + (−0.707 − 0.707i)18-s + i·19-s − 1.73·21-s − i·24-s + (−0.707 + 0.707i)27-s − 1.73i·29-s + (−0.707 − 0.707i)38-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s − 1.00·6-s + (−1.22 + 1.22i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s − 1.73i·14-s + 1.00·16-s + (−0.707 − 0.707i)18-s + i·19-s − 1.73·21-s − i·24-s + (−0.707 + 0.707i)27-s − 1.73i·29-s + (−0.707 − 0.707i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1425\)    =    \(3 \cdot 5^{2} \cdot 19\)
Sign: $-0.945 + 0.326i$
Analytic conductor: \(0.711167\)
Root analytic conductor: \(0.843307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1425} (968, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1425,\ (\ :0),\ -0.945 + 0.326i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6430970722\)
\(L(\frac12)\) \(\approx\) \(0.6430970722\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 \)
19 \( 1 - iT \)
good2 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
7 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + 1.73iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + 1.73T + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
59 \( 1 - 1.73iT - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 1.73T + T^{2} \)
73 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - 1.73iT - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.658208665004911117207361849298, −9.442087238422849462571634782075, −8.493220296441734375200923824700, −8.122811790030963499725550841337, −7.06750450824806579504866072291, −6.16530620142373789953552377171, −5.48943735445424056919466304459, −4.04195448403740388037631866470, −3.24271011178538409171159418243, −2.36667364711556143619609608140, 0.59167177883557388024564707434, 1.80024048932092364627599612678, 3.03056216960866377117179366632, 3.59223016141743187437532983317, 5.07173724360936179741679674889, 6.46694933084267962225753497477, 6.84654938424630517759177553754, 7.78994652114589879742154845747, 8.750192083732866016092214994003, 9.325567771850745694189639707708

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