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Properties

Label 2-1425-285.149-c0-0-1
Degree $2$
Conductor $1425$
Sign $0.923 + 0.383i$
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  + (0.342 + 0.939i)3-s + (−0.173 − 0.984i)4-s + (1.32 − 0.766i)7-s + (−0.766 + 0.642i)9-s + (0.866 − 0.5i)12-s + (0.642 − 1.76i)13-s + (−0.939 + 0.342i)16-s − 19-s + (1.17 + 0.984i)21-s + (−0.866 − 0.500i)27-s + (−0.984 − 1.17i)28-s + (−0.173 − 0.300i)31-s + (0.766 + 0.642i)36-s + 1.53i·37-s + 1.87·39-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)3-s + (−0.173 − 0.984i)4-s + (1.32 − 0.766i)7-s + (−0.766 + 0.642i)9-s + (0.866 − 0.5i)12-s + (0.642 − 1.76i)13-s + (−0.939 + 0.342i)16-s − 19-s + (1.17 + 0.984i)21-s + (−0.866 − 0.500i)27-s + (−0.984 − 1.17i)28-s + (−0.173 − 0.300i)31-s + (0.766 + 0.642i)36-s + 1.53i·37-s + 1.87·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1425\)    =    \(3 \cdot 5^{2} \cdot 19\)
Sign: $0.923 + 0.383i$
Analytic conductor: \(0.711167\)
Root analytic conductor: \(0.843307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1425} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1425,\ (\ :0),\ 0.923 + 0.383i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.289448550\)
\(L(\frac12)\) \(\approx\) \(1.289448550\)
\(L(1)\) 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 \( 1 + T \)
good2 \( 1 + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (-1.32 + 0.766i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.642 + 1.76i)T + (-0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.173 + 0.984i)T^{2} \)
23 \( 1 + (-0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - 1.53iT - T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (-1.85 - 0.326i)T + (0.939 + 0.342i)T^{2} \)
47 \( 1 + (0.173 - 0.984i)T^{2} \)
53 \( 1 + (-0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (-0.223 - 0.266i)T + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-0.118 - 0.326i)T + (-0.766 + 0.642i)T^{2} \)
79 \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.766 - 0.642i)T^{2} \)
97 \( 1 + (-0.642 + 0.766i)T + (-0.173 - 0.984i)T^{2} \)
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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

−9.951031388203659479647415050830, −8.811592276102201786799494536638, −8.277616417483653464195413261884, −7.49889616360573942515014402925, −6.10692170557083598555366417447, −5.39139280947006171747344930594, −4.62982346198203930501050630781, −3.92732261670760262799868793213, −2.58234361727971328639299373687, −1.16977377409637096445597600895, 1.78739416163968927083805523198, 2.40431795525876931228397325740, 3.79326287740613596802343478927, 4.57680988653292289926820658736, 5.80254919176894349916390669421, 6.72545146721745990192899685648, 7.48849127150112623977900675723, 8.227722059626154872372445121732, 8.872959308688115503655357420316, 9.190173935719075480552528696523

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