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Properties

Label 2-1500-25.4-c1-0-1
Degree $2$
Conductor $1500$
Sign $-0.789 - 0.614i$
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 + 0.747i·7-s + (−0.309 − 0.951i)9-s + (0.0646 − 0.198i)11-s + (−2.38 + 0.773i)13-s + (4.00 + 5.51i)17-s + (1.00 − 0.731i)19-s + (−0.604 − 0.439i)21-s + (−3.09 − 1.00i)23-s + (0.951 + 0.309i)27-s + (−4.19 − 3.04i)29-s + (−3.02 + 2.19i)31-s + (0.122 + 0.169i)33-s + (−1.86 + 0.607i)37-s + (0.773 − 2.38i)39-s + ⋯
L(s)  = 1  + (−0.339 + 0.467i)3-s + 0.282i·7-s + (−0.103 − 0.317i)9-s + (0.0194 − 0.0599i)11-s + (−0.660 + 0.214i)13-s + (0.972 + 1.33i)17-s + (0.231 − 0.167i)19-s + (−0.131 − 0.0958i)21-s + (−0.644 − 0.209i)23-s + (0.183 + 0.0594i)27-s + (−0.778 − 0.565i)29-s + (−0.543 + 0.394i)31-s + (0.0213 + 0.0294i)33-s + (−0.307 + 0.0998i)37-s + (0.123 − 0.381i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.789 - 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.789 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.789 - 0.614i$
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: \((2,\ 1500,\ (\ :1/2),\ -0.789 - 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8346949381\)
\(L(\frac12)\) \(\approx\) \(0.8346949381\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.747iT - 7T^{2} \)
11 \( 1 + (-0.0646 + 0.198i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (2.38 - 0.773i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-4.00 - 5.51i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.00 + 0.731i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (3.09 + 1.00i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (4.19 + 3.04i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.02 - 2.19i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.86 - 0.607i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.993 - 3.05i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 12.7iT - 43T^{2} \)
47 \( 1 + (3.81 - 5.24i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.43 - 3.35i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.61 + 11.1i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (3.85 - 11.8i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (1.71 + 2.35i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (5.29 + 3.85i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.39 - 0.778i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (8.28 + 6.02i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.33 + 4.59i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-0.284 + 0.876i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (9.13 - 12.5i)T + (-29.9 - 92.2i)T^{2} \)
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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

−9.791437102392575504611060126999, −9.173969890806096197175449052034, −8.158446757346523456649464368250, −7.50780868915643756103018157282, −6.31646640664413372552997286430, −5.75510528627549916077660009893, −4.79015095366202531538535418694, −3.92438405677084074845319892498, −2.90282304974166548734056748965, −1.54176053331056054403811180671, 0.34271175110074708263497172648, 1.77225293920801966791343258521, 2.97001416705043633818071314606, 4.04541811444617462010314336881, 5.27752806305118845844358496093, 5.66738377744390554087669242259, 7.08697123215814349020065569998, 7.28310348273657585980610521353, 8.229954139169140677855873271855, 9.267782533432351731575435054542

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