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Properties

Label 2-1725-345.68-c0-0-7
Degree $2$
Conductor $1725$
Sign $-0.0438 - 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0438 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0438 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1725\)    =    \(3 \cdot 5^{2} \cdot 23\)
Sign: $-0.0438 - 0.999i$
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: \((2,\ 1725,\ (\ :0),\ -0.0438 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.253106802\)
\(L(\frac12)\) \(\approx\) \(2.253106802\)
\(L(1)\) 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

−9.011288856435008400974068618889, −8.721669024460518908617525628568, −7.79269160757003083036911624448, −7.02489775900263883081572645037, −6.53323366036371377594955570720, −5.89017724967890002255752175324, −5.01279081823274035153579394587, −3.98939185912251884155484080773, −3.22272117669523268982206263901, −1.91304319798826999374970990231, 1.31371106965322872681505555684, 2.81448737266651728419129907381, 3.37013664662342081781124415056, 4.08838425127025574833803138456, 5.04873093088278572601802045250, 5.62300016268065515212992679489, 6.32807305847412322108177450349, 7.88785129335326860897856339236, 8.944853026529841742124789249365, 9.610054805305178593552075074093

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