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Properties

Label 2-1725-345.137-c0-0-6
Degree $2$
Conductor $1725$
Sign $-0.300 - 0.953i$
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  + (0.245 − 0.245i)2-s + (0.422 + 0.906i)3-s + 0.879i·4-s + (0.326 + 0.118i)6-s + (0.461 + 0.461i)8-s + (−0.642 + 0.766i)9-s + (−0.796 + 0.371i)12-s + (−0.909 + 0.909i)13-s − 0.652·16-s + (0.0302 + 0.345i)18-s + (−0.707 − 0.707i)23-s + (−0.223 + 0.613i)24-s + 0.446i·26-s + (−0.965 − 0.258i)27-s + 1.96·29-s + ⋯
L(s)  = 1  + (0.245 − 0.245i)2-s + (0.422 + 0.906i)3-s + 0.879i·4-s + (0.326 + 0.118i)6-s + (0.461 + 0.461i)8-s + (−0.642 + 0.766i)9-s + (−0.796 + 0.371i)12-s + (−0.909 + 0.909i)13-s − 0.652·16-s + (0.0302 + 0.345i)18-s + (−0.707 − 0.707i)23-s + (−0.223 + 0.613i)24-s + 0.446i·26-s + (−0.965 − 0.258i)27-s + 1.96·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1725\)    =    \(3 \cdot 5^{2} \cdot 23\)
Sign: $-0.300 - 0.953i$
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: \((2,\ 1725,\ (\ :0),\ -0.300 - 0.953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.341515793\)
\(L(\frac12)\) \(\approx\) \(1.341515793\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.422 - 0.906i)T \)
5 \( 1 \)
23 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (-0.245 + 0.245i)T - iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (0.909 - 0.909i)T - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
29 \( 1 - 1.96T + T^{2} \)
31 \( 1 - 0.347T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + 0.684iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-1.08 + 1.08i)T - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - 1.73T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - 1.28iT - T^{2} \)
73 \( 1 + (1.39 - 1.39i)T - iT^{2} \)
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

−9.858464437746934836428891176208, −8.773125328662826957102759640147, −8.456666275991575418195796974531, −7.47465545789465013062835118534, −6.68805318452882598410373631383, −5.40247211246060421071075707928, −4.43989992541522783424739987856, −4.07968386085024458279161423459, −2.87622562287272406185040660507, −2.25846345160483832085855037261, 0.908936165133460936968583623274, 2.14929752434623699269427442878, 3.11336874043521160800842851207, 4.44089920818776276825191580465, 5.37589865170629576154392300938, 6.11873431022282621193665471841, 6.84535164053255295754212983278, 7.63111925181520640030312565655, 8.313670601576108843052614365257, 9.294581611815266922906845189360

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