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Properties

Label 2-1500-25.4-c1-0-4
Degree $2$
Conductor $1500$
Sign $0.882 - 0.469i$
Analytic cond. $11.9775$
Root an. cond. $3.46086$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.587 + 0.809i)3-s + 0.0883i·7-s + (−0.309 − 0.951i)9-s + (0.701 − 2.15i)11-s + (−2.52 + 0.819i)13-s + (1.22 + 1.68i)17-s + (1.42 − 1.03i)19-s + (−0.0714 − 0.0519i)21-s + (4.50 + 1.46i)23-s + (0.951 + 0.309i)27-s + (2.99 + 2.17i)29-s + (3.32 − 2.41i)31-s + (1.33 + 1.83i)33-s + (6.77 − 2.19i)37-s + (0.819 − 2.52i)39-s + ⋯
L(s)  = 1  + (−0.339 + 0.467i)3-s + 0.0333i·7-s + (−0.103 − 0.317i)9-s + (0.211 − 0.650i)11-s + (−0.699 + 0.227i)13-s + (0.297 + 0.409i)17-s + (0.326 − 0.237i)19-s + (−0.0155 − 0.0113i)21-s + (0.940 + 0.305i)23-s + (0.183 + 0.0594i)27-s + (0.556 + 0.404i)29-s + (0.596 − 0.433i)31-s + (0.232 + 0.319i)33-s + (1.11 − 0.361i)37-s + (0.131 − 0.403i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.469i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $0.882 - 0.469i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ 0.882 - 0.469i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.466219396\)
\(L(\frac12)\) \(\approx\) \(1.466219396\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 \)
good7 \( 1 - 0.0883iT - 7T^{2} \)
11 \( 1 + (-0.701 + 2.15i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (2.52 - 0.819i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.22 - 1.68i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.42 + 1.03i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-4.50 - 1.46i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-2.99 - 2.17i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-3.32 + 2.41i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-6.77 + 2.19i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.03 + 6.26i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.79iT - 43T^{2} \)
47 \( 1 + (5.94 - 8.17i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.565 - 0.777i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.77 - 8.53i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.92 + 9.00i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-8.17 - 11.2i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (4.97 + 3.61i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.00 + 0.975i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-10.8 - 7.84i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-9.29 - 12.7i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-3.16 + 9.74i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-7.52 + 10.3i)T + (-29.9 - 92.2i)T^{2} \)
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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

−9.541754845410479367567652311310, −8.895557437857588139941668015737, −7.978451775242793276511894441562, −7.07376928903496702878285535126, −6.20703755582757560420708784698, −5.37730864260058184255912495336, −4.56617570164379233017883243338, −3.57824299581952833546246785383, −2.57217569109772065751826036617, −0.941123268688664072535361181396, 0.848600309139395157518466889791, 2.19755201768859503058957129233, 3.23415414420912391627685245662, 4.59903386864075269717982124646, 5.18522074674013892215948290793, 6.29576821589790911382190771389, 6.99189196901303886116167156661, 7.71506625118539139755347125429, 8.512347880433438202295015316481, 9.579896859259370490241236858647

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