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Properties

Label 2-1725-345.68-c0-0-3
Degree $2$
Conductor $1725$
Sign $0.887 + 0.461i$
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 + (0.819 + 0.573i)3-s + 1.34i·4-s + (−0.266 − 1.50i)6-s + (0.376 − 0.376i)8-s + (0.342 + 0.939i)9-s + (−0.772 + 1.10i)12-s + (−0.483 − 0.483i)13-s + 0.532·16-s + (0.647 − 1.38i)18-s + (0.707 − 0.707i)23-s + (0.524 − 0.0923i)24-s + 1.04i·26-s + (−0.258 + 0.965i)27-s + 1.28·29-s + ⋯
L(s)  = 1  + (−1.08 − 1.08i)2-s + (0.819 + 0.573i)3-s + 1.34i·4-s + (−0.266 − 1.50i)6-s + (0.376 − 0.376i)8-s + (0.342 + 0.939i)9-s + (−0.772 + 1.10i)12-s + (−0.483 − 0.483i)13-s + 0.532·16-s + (0.647 − 1.38i)18-s + (0.707 − 0.707i)23-s + (0.524 − 0.0923i)24-s + 1.04i·26-s + (−0.258 + 0.965i)27-s + 1.28·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1725\)    =    \(3 \cdot 5^{2} \cdot 23\)
Sign: $0.887 + 0.461i$
Analytic conductor: \(0.860887\)
Root analytic conductor: \(0.927840\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1725} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1725,\ (\ :0),\ 0.887 + 0.461i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8516087610\)
\(L(\frac12)\) \(\approx\) \(0.8516087610\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.819 - 0.573i)T \)
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

−9.610526267820456193384716625033, −8.795294374441430963657756888606, −8.230616457109572335443514999703, −7.61089068271224194687609845917, −6.41543256694776052566753869750, −5.06097752071792132925886135202, −4.22498551560566832327698144964, −2.90749227895365978601425637112, −2.67582326679568882382673623829, −1.23550940524126955659640863750, 1.04940749209577766311282084232, 2.40242659267476443135336011405, 3.55042803820947300462959710851, 4.81961817489826159655391665941, 6.01051198990078101572592482623, 6.71784022886458431329194208068, 7.34998478218364138323413172765, 7.935594941940235514081513680274, 8.771850874681024013251789190281, 9.170382203417289396745466629175

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