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Properties

Label 2-1500-25.9-c1-0-2
Degree $2$
Conductor $1500$
Sign $0.0191 - 0.999i$
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  + (0.951 − 0.309i)3-s + 4.62i·7-s + (0.809 − 0.587i)9-s + (4.00 + 2.90i)11-s + (−2.21 − 3.04i)13-s + (−2.55 − 0.831i)17-s + (−1.81 + 5.58i)19-s + (1.42 + 4.40i)21-s + (−3.92 + 5.40i)23-s + (0.587 − 0.809i)27-s + (−0.370 − 1.14i)29-s + (1.02 − 3.14i)31-s + (4.70 + 1.52i)33-s + (−1.10 − 1.51i)37-s + (−3.04 − 2.21i)39-s + ⋯
L(s)  = 1  + (0.549 − 0.178i)3-s + 1.74i·7-s + (0.269 − 0.195i)9-s + (1.20 + 0.877i)11-s + (−0.613 − 0.844i)13-s + (−0.620 − 0.201i)17-s + (−0.416 + 1.28i)19-s + (0.311 + 0.960i)21-s + (−0.818 + 1.12i)23-s + (0.113 − 0.155i)27-s + (−0.0688 − 0.212i)29-s + (0.183 − 0.564i)31-s + (0.819 + 0.266i)33-s + (−0.181 − 0.249i)37-s + (−0.487 − 0.354i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0191 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0191 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $0.0191 - 0.999i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ 0.0191 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.798568449\)
\(L(\frac12)\) \(\approx\) \(1.798568449\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 \)
good7 \( 1 - 4.62iT - 7T^{2} \)
11 \( 1 + (-4.00 - 2.90i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (2.21 + 3.04i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.55 + 0.831i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.81 - 5.58i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (3.92 - 5.40i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.370 + 1.14i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.02 + 3.14i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.10 + 1.51i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.45 + 1.78i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 10.6iT - 43T^{2} \)
47 \( 1 + (0.246 - 0.0801i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-9.31 + 3.02i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (7.78 - 5.65i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (5.07 + 3.68i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-2.43 - 0.791i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.68 - 8.25i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.86 - 3.94i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.85 + 11.8i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-8.45 - 2.74i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-11.7 - 8.56i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-3.79 + 1.23i)T + (78.4 - 57.0i)T^{2} \)
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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

−9.516939470481778278346111753141, −8.970500419193029630654698962238, −8.118075991745263608774817000002, −7.44257432712577416846094950459, −6.29019002759577120146480508149, −5.73292897040138622273195001177, −4.62764243708084963557845074041, −3.59453050404128532060646180978, −2.45905207064811023689996964310, −1.74463618360702720299762080477, 0.65220819882669447880333800158, 2.03463950729633410885301169793, 3.38893378689343480166369233920, 4.20061598860232695855339057443, 4.68800995958704932480322621639, 6.37651467733183122129828595373, 6.84446035307518580313573637316, 7.60364348204378929565991468062, 8.709050343991360491812951998624, 9.095119286701030493645535907529

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