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Properties

Label 2-1500-25.19-c1-0-0
Degree $2$
Conductor $1500$
Sign $-0.468 - 0.883i$
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 − 4.41i·7-s + (−0.309 + 0.951i)9-s + (1.37 + 4.23i)11-s + (−5.46 − 1.77i)13-s + (−3.86 + 5.31i)17-s + (−2.25 − 1.63i)19-s + (−3.57 + 2.59i)21-s + (−1.25 + 0.406i)23-s + (0.951 − 0.309i)27-s + (−3.91 + 2.84i)29-s + (0.159 + 0.115i)31-s + (2.61 − 3.60i)33-s + (7.80 + 2.53i)37-s + (1.77 + 5.46i)39-s + ⋯
L(s)  = 1  + (−0.339 − 0.467i)3-s − 1.66i·7-s + (−0.103 + 0.317i)9-s + (0.414 + 1.27i)11-s + (−1.51 − 0.492i)13-s + (−0.936 + 1.28i)17-s + (−0.516 − 0.375i)19-s + (−0.779 + 0.566i)21-s + (−0.260 + 0.0847i)23-s + (0.183 − 0.0594i)27-s + (−0.727 + 0.528i)29-s + (0.0286 + 0.0208i)31-s + (0.455 − 0.626i)33-s + (1.28 + 0.417i)37-s + (0.284 + 0.875i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.468 - 0.883i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ -0.468 - 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2148351235\)
\(L(\frac12)\) \(\approx\) \(0.2148351235\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.41iT - 7T^{2} \)
11 \( 1 + (-1.37 - 4.23i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (5.46 + 1.77i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (3.86 - 5.31i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.25 + 1.63i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.25 - 0.406i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (3.91 - 2.84i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.159 - 0.115i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-7.80 - 2.53i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.42 + 7.46i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 0.412iT - 43T^{2} \)
47 \( 1 + (-4.58 - 6.30i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.185 - 0.255i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.778 - 2.39i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.88 - 8.87i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (7.02 - 9.66i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (0.411 - 0.299i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (14.9 - 4.87i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.77 - 2.01i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.15 - 4.34i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (3.50 + 10.7i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (2.98 + 4.10i)T + (-29.9 + 92.2i)T^{2} \)
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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

−9.998727767294459389721241326025, −8.981648320169514122128384251688, −7.81970422995971987876529255118, −7.23265969182984995821650764092, −6.81582958302940553759143227524, −5.70554435695716511272469511303, −4.40859682088677526468242272480, −4.20308052947517656543697822556, −2.52647922946733637210640194091, −1.41181728366743937004055795243, 0.086287120488983424786271199200, 2.20062786979477940413299684363, 2.93628398312797916403535948947, 4.28598664313795619479510583773, 5.11030977898889955477572546289, 5.89225586631544933030074775881, 6.53507945687195232780248052177, 7.69621652341721974118089333115, 8.669986826954543289040167117111, 9.284219855033459760238122394795

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