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Properties

Label 2-1500-25.9-c1-0-13
Degree $2$
Conductor $1500$
Sign $-0.580 + 0.814i$
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 − 4.78i·7-s + (0.809 − 0.587i)9-s + (−1.58 − 1.14i)11-s + (0.634 + 0.873i)13-s + (−3.61 − 1.17i)17-s + (−1.31 + 4.04i)19-s + (−1.47 − 4.54i)21-s + (3.44 − 4.74i)23-s + (0.587 − 0.809i)27-s + (−3.26 − 10.0i)29-s + (−1.33 + 4.10i)31-s + (−1.86 − 0.604i)33-s + (−3.32 − 4.57i)37-s + (0.873 + 0.634i)39-s + ⋯
L(s)  = 1  + (0.549 − 0.178i)3-s − 1.80i·7-s + (0.269 − 0.195i)9-s + (−0.477 − 0.346i)11-s + (0.176 + 0.242i)13-s + (−0.877 − 0.285i)17-s + (−0.301 + 0.927i)19-s + (−0.322 − 0.992i)21-s + (0.718 − 0.989i)23-s + (0.113 − 0.155i)27-s + (−0.605 − 1.86i)29-s + (−0.239 + 0.737i)31-s + (−0.323 − 0.105i)33-s + (−0.546 − 0.752i)37-s + (0.139 + 0.101i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.580 + 0.814i$
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: \((2,\ 1500,\ (\ :1/2),\ -0.580 + 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.507491522\)
\(L(\frac12)\) \(\approx\) \(1.507491522\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.78iT - 7T^{2} \)
11 \( 1 + (1.58 + 1.14i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.634 - 0.873i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.61 + 1.17i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.31 - 4.04i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-3.44 + 4.74i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (3.26 + 10.0i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.33 - 4.10i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.32 + 4.57i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (0.694 - 0.504i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 10.8iT - 43T^{2} \)
47 \( 1 + (2.85 - 0.927i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.01 + 1.30i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.85 - 2.80i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.93 + 2.13i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-6.59 - 2.14i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (3.70 + 11.4i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-9.96 + 13.7i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.04 - 6.29i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.45 - 0.797i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (0.673 + 0.489i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-8.67 + 2.81i)T + (78.4 - 57.0i)T^{2} \)
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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

−9.216669825758092498164605051666, −8.228047699020805575839023669702, −7.67989618197085936505891581163, −6.87538881044821311819519402528, −6.16975535715588053909764338685, −4.73382061299792605734607541759, −4.07330458113015664110877947894, −3.17986649376431996474628991137, −1.88893953713982986954181603973, −0.52785959472297253035564388897, 1.89077616318829070906006933014, 2.65867198772805581846797903293, 3.60724436514734283921049395258, 4.99901498561636747865469079049, 5.41541840704369954715301633241, 6.55796180719157179320425055540, 7.39487814097233723945132737412, 8.487951540838729572516409749145, 8.900982943230808291779073320374, 9.462624259250629774389953156049

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