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Properties

Label 2-1725-345.137-c0-0-1
Degree $2$
Conductor $1725$
Sign $-0.970 - 0.242i$
Analytic cond. $0.860887$
Root an. cond. $0.927840$
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  + (−1.32 + 1.32i)2-s + (−0.0871 − 0.996i)3-s − 2.53i·4-s + (1.43 + 1.20i)6-s + (2.03 + 2.03i)8-s + (−0.984 + 0.173i)9-s + (−2.52 + 0.220i)12-s + (−1.39 + 1.39i)13-s − 2.87·16-s + (1.07 − 1.53i)18-s + (−0.707 − 0.707i)23-s + (1.85 − 2.20i)24-s − 3.70i·26-s + (0.258 + 0.965i)27-s + 0.684·29-s + ⋯
L(s)  = 1  + (−1.32 + 1.32i)2-s + (−0.0871 − 0.996i)3-s − 2.53i·4-s + (1.43 + 1.20i)6-s + (2.03 + 2.03i)8-s + (−0.984 + 0.173i)9-s + (−2.52 + 0.220i)12-s + (−1.39 + 1.39i)13-s − 2.87·16-s + (1.07 − 1.53i)18-s + (−0.707 − 0.707i)23-s + (1.85 − 2.20i)24-s − 3.70i·26-s + (0.258 + 0.965i)27-s + 0.684·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1725\)    =    \(3 \cdot 5^{2} \cdot 23\)
Sign: $-0.970 - 0.242i$
Analytic conductor: \(0.860887\)
Root analytic conductor: \(0.927840\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1725} (482, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1725,\ (\ :0),\ -0.970 - 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1743161595\)
\(L(\frac12)\) \(\approx\) \(0.1743161595\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0871 + 0.996i)T \)
5 \( 1 \)
23 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (1.32 - 1.32i)T - iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (1.39 - 1.39i)T - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
29 \( 1 - 0.684T + T^{2} \)
31 \( 1 + 1.87T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - 1.28iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.245 + 0.245i)T - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + 1.73T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - 1.96iT - T^{2} \)
73 \( 1 + (0.483 - 0.483i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
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

−9.459700365586996764381486430829, −8.981850748172334097944701713422, −8.105249163833114595469804271130, −7.48938361068847792699860022872, −6.87989869195613923088769093611, −6.31328168679542663349606619009, −5.44632332218460766209123601473, −4.51520184458134635644836423014, −2.43308596367019534623189317793, −1.47364292546457334201517376529, 0.20380668910147702764961883413, 2.02699394460297100304066480539, 3.01886020266473214281181321276, 3.66876300276711372762164561766, 4.77565136299574748755104128149, 5.72970852325117639140706937610, 7.30348421013736339538781707053, 7.86821961708338816362899389581, 8.721559590061292314986505527652, 9.418655763798456500578061244646

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