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Properties

Label 2-1725-345.137-c0-0-3
Degree $2$
Conductor $1725$
Sign $-0.587 - 0.809i$
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  + (−1.08 + 1.08i)2-s + (−0.573 + 0.819i)3-s − 1.34i·4-s + (−0.266 − 1.50i)6-s + (0.376 + 0.376i)8-s + (−0.342 − 0.939i)9-s + (1.10 + 0.772i)12-s + (0.483 − 0.483i)13-s + 0.532·16-s + (1.38 + 0.647i)18-s + (0.707 + 0.707i)23-s + (−0.524 + 0.0923i)24-s + 1.04i·26-s + (0.965 + 0.258i)27-s − 1.28·29-s + ⋯
L(s)  = 1  + (−1.08 + 1.08i)2-s + (−0.573 + 0.819i)3-s − 1.34i·4-s + (−0.266 − 1.50i)6-s + (0.376 + 0.376i)8-s + (−0.342 − 0.939i)9-s + (1.10 + 0.772i)12-s + (0.483 − 0.483i)13-s + 0.532·16-s + (1.38 + 0.647i)18-s + (0.707 + 0.707i)23-s + (−0.524 + 0.0923i)24-s + 1.04i·26-s + (0.965 + 0.258i)27-s − 1.28·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1725\)    =    \(3 \cdot 5^{2} \cdot 23\)
Sign: $-0.587 - 0.809i$
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: \((2,\ 1725,\ (\ :0),\ -0.587 - 0.809i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4989096115\)
\(L(\frac12)\) \(\approx\) \(0.4989096115\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.573 - 0.819i)T \)
5 \( 1 \)
23 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (1.08 - 1.08i)T - iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-0.483 + 0.483i)T - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
29 \( 1 + 1.28T + T^{2} \)
31 \( 1 - 1.53T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - 1.96iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-1.32 + 1.32i)T - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - 1.73T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - 0.684iT - T^{2} \)
73 \( 1 + (0.909 - 0.909i)T - iT^{2} \)
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

−9.747466048413068846226828111447, −8.924495410609519301722734720424, −8.343923160089566353298724393147, −7.43832197008040677888070734315, −6.64645529561306378767071148679, −5.86362958448700963021962435001, −5.28104310388019080038564664308, −4.13109630408469677655930732172, −3.07141622891403795681037575536, −1.02404571961761974461252590283, 0.78534950535080911868456228677, 1.88842193452938431891570543022, 2.71765321846907560677040876661, 3.96985442747801107311882917455, 5.24604377752460905739762367597, 6.15341866632312990525929600441, 7.05724751407479150464725766010, 7.82272041754482216448329468271, 8.660340191836063680952875132011, 9.167061597716660319093829338790

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