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Properties

Label 2-2535-195.134-c0-0-1
Degree $2$
Conductor $2535$
Sign $-0.890 - 0.455i$
Analytic cond. $1.26512$
Root an. cond. $1.12477$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.385 + 0.222i)2-s + (−0.5 + 0.866i)3-s + (−0.400 − 0.694i)4-s + i·5-s + (−0.385 + 0.222i)6-s − 0.801i·8-s + (−0.499 − 0.866i)9-s + (−0.222 + 0.385i)10-s + 0.801·12-s + (−0.866 − 0.5i)15-s + (−0.222 + 0.385i)16-s + (0.623 + 1.07i)17-s − 0.445i·18-s + (−1.56 + 0.900i)19-s + (0.694 − 0.400i)20-s + ⋯
L(s)  = 1  + (0.385 + 0.222i)2-s + (−0.5 + 0.866i)3-s + (−0.400 − 0.694i)4-s + i·5-s + (−0.385 + 0.222i)6-s − 0.801i·8-s + (−0.499 − 0.866i)9-s + (−0.222 + 0.385i)10-s + 0.801·12-s + (−0.866 − 0.5i)15-s + (−0.222 + 0.385i)16-s + (0.623 + 1.07i)17-s − 0.445i·18-s + (−1.56 + 0.900i)19-s + (0.694 − 0.400i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2535\)    =    \(3 \cdot 5 \cdot 13^{2}\)
Sign: $-0.890 - 0.455i$
Analytic conductor: \(1.26512\)
Root analytic conductor: \(1.12477\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2535} (1499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2535,\ (\ :0),\ -0.890 - 0.455i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7058344658\)
\(L(\frac12)\) \(\approx\) \(0.7058344658\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 - iT \)
13 \( 1 \)
good2 \( 1 + (-0.385 - 0.222i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.56 - 0.900i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - 1.24iT - T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 - 1.80iT - T^{2} \)
53 \( 1 + 1.80T + T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 0.445T + T^{2} \)
83 \( 1 - 1.24iT - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−9.693711964734581638997584497857, −8.758899514031349295575509812316, −7.912260648071154891698956239468, −6.69127987727629731402212601467, −6.18140415608530103862440430859, −5.65632635549398632785178223337, −4.65901187194940210016059310662, −3.92382617215348261063208293674, −3.24019436415551651408764619128, −1.68015380610804058975627226299, 0.43043086973810315475576897111, 1.96919054522780820460091146803, 2.85108353281014871145832841024, 4.17769160887456357074280865884, 4.78006715032623820248184301200, 5.51210339490929861447512511031, 6.38587546609281023771672457777, 7.38829612542297214550923642246, 7.956643246738809583475095876278, 8.696964373481688840863318082776

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