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Properties

Label 2-1425-57.35-c0-0-0
Degree $2$
Conductor $1425$
Sign $-0.873 - 0.486i$
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  + (−0.223 + 0.266i)2-s + (0.342 + 0.939i)3-s + (0.152 + 0.866i)4-s + (−0.326 − 0.118i)6-s + (−0.565 − 0.326i)8-s + (−0.766 + 0.642i)9-s + (−0.761 + 0.439i)12-s + (−0.613 + 0.223i)16-s + (−0.984 + 1.17i)17-s − 0.347i·18-s + (0.939 − 0.342i)19-s + (−0.984 + 0.173i)23-s + (0.113 − 0.642i)24-s + (−0.866 − 0.500i)27-s + (0.939 + 1.62i)31-s + (0.300 − 0.826i)32-s + ⋯
L(s)  = 1  + (−0.223 + 0.266i)2-s + (0.342 + 0.939i)3-s + (0.152 + 0.866i)4-s + (−0.326 − 0.118i)6-s + (−0.565 − 0.326i)8-s + (−0.766 + 0.642i)9-s + (−0.761 + 0.439i)12-s + (−0.613 + 0.223i)16-s + (−0.984 + 1.17i)17-s − 0.347i·18-s + (0.939 − 0.342i)19-s + (−0.984 + 0.173i)23-s + (0.113 − 0.642i)24-s + (−0.866 − 0.500i)27-s + (0.939 + 1.62i)31-s + (0.300 − 0.826i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1425\)    =    \(3 \cdot 5^{2} \cdot 19\)
Sign: $-0.873 - 0.486i$
Analytic conductor: \(0.711167\)
Root analytic conductor: \(0.843307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1425} (776, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1425,\ (\ :0),\ -0.873 - 0.486i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9597215939\)
\(L(\frac12)\) \(\approx\) \(0.9597215939\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.342 - 0.939i)T \)
5 \( 1 \)
19 \( 1 + (-0.939 + 0.342i)T \)
good2 \( 1 + (0.223 - 0.266i)T + (-0.173 - 0.984i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.766 + 0.642i)T^{2} \)
17 \( 1 + (0.984 - 1.17i)T + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.984 - 0.173i)T + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.223 - 0.266i)T + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (-1.50 + 0.266i)T + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (0.766 - 0.642i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
83 \( 1 + (-1.62 + 0.939i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.766 - 0.642i)T^{2} \)
97 \( 1 + (0.173 + 0.984i)T^{2} \)
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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

−10.01732434288412670583435265376, −9.060661122177109341065211936142, −8.552115205328903379164240566777, −7.88447633137835930157819200235, −6.93087868371700578115633648262, −6.03175118954456792844807236686, −4.91507946126097904883509965007, −4.00199105813083000859840600946, −3.28892564739368982030107174492, −2.22552331280362655659112178152, 0.800278491991428367376784376059, 2.10015384953825829591621243461, 2.81268197482395631362226345471, 4.27517023626537210311764032040, 5.49472275986230981645659315977, 6.15142649304796589925211364279, 7.00106740396581708999454323846, 7.74163951586264483845769396149, 8.702303188450885682428020334708, 9.407308749659007501529561438493

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