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Properties

Label 2-1425-57.11-c0-0-1
Degree $2$
Conductor $1425$
Sign $0.740 - 0.671i$
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  + (0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 + 0.866i)6-s − i·8-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + 0.999i·18-s + (0.5 − 0.866i)19-s + (−1.73 + i)23-s + (0.499 − 0.866i)24-s + 0.999i·27-s − 31-s + (0.499 + 0.866i)34-s + (0.866 − 0.499i)38-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 + 0.866i)6-s − i·8-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + 0.999i·18-s + (0.5 − 0.866i)19-s + (−1.73 + i)23-s + (0.499 − 0.866i)24-s + 0.999i·27-s − 31-s + (0.499 + 0.866i)34-s + (0.866 − 0.499i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1425\)    =    \(3 \cdot 5^{2} \cdot 19\)
Sign: $0.740 - 0.671i$
Analytic conductor: \(0.711167\)
Root analytic conductor: \(0.843307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1425} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1425,\ (\ :0),\ 0.740 - 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.121846650\)
\(L(\frac12)\) \(\approx\) \(2.121846650\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
19 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - iT - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−9.716303966951986751022077690045, −9.187207937445612669602314057552, −8.031958628352518892065376764289, −7.48585266497365640419109323147, −6.42711092465672408619745207976, −5.51522329199726112390654103173, −4.81529347293065862248146364008, −3.83595549117413387555434995983, −3.25998723815707987075383792916, −1.76221632613954263932862067452, 1.69416231243050336798843588352, 2.68955634152112399463737349430, 3.55553032289690054234387556431, 4.22872399483125740436786220248, 5.38518729482014767113970436533, 6.23798308642972134082951922247, 7.42528488671003018995808636602, 8.017523154318050638031613969451, 8.699524744092882179066266324919, 9.677414151667256226864334983848

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