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Properties

Label 2-30e2-300.83-c0-0-1
Degree $2$
Conductor $900$
Sign $-0.762 + 0.647i$
Analytic cond. $0.449158$
Root an. cond. $0.670192$
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  + (0.453 − 0.891i)2-s + (−0.587 − 0.809i)4-s + (−0.156 − 0.987i)5-s + (−0.987 + 0.156i)8-s + (−0.951 − 0.309i)10-s + (0.278 − 0.142i)13-s + (−0.309 + 0.951i)16-s + (−0.297 − 1.87i)17-s + (−0.707 + 0.707i)20-s + (−0.951 + 0.309i)25-s − 0.312i·26-s + (−0.734 + 0.533i)29-s + (0.707 + 0.707i)32-s + (−1.80 − 0.587i)34-s + (0.809 + 1.58i)37-s + ⋯
L(s)  = 1  + (0.453 − 0.891i)2-s + (−0.587 − 0.809i)4-s + (−0.156 − 0.987i)5-s + (−0.987 + 0.156i)8-s + (−0.951 − 0.309i)10-s + (0.278 − 0.142i)13-s + (−0.309 + 0.951i)16-s + (−0.297 − 1.87i)17-s + (−0.707 + 0.707i)20-s + (−0.951 + 0.309i)25-s − 0.312i·26-s + (−0.734 + 0.533i)29-s + (0.707 + 0.707i)32-s + (−1.80 − 0.587i)34-s + (0.809 + 1.58i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.762 + 0.647i$
Analytic conductor: \(0.449158\)
Root analytic conductor: \(0.670192\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :0),\ -0.762 + 0.647i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.064468794\)
\(L(\frac12)\) \(\approx\) \(1.064468794\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.453 + 0.891i)T \)
3 \( 1 \)
5 \( 1 + (0.156 + 0.987i)T \)
good7 \( 1 + iT^{2} \)
11 \( 1 + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.278 + 0.142i)T + (0.587 - 0.809i)T^{2} \)
17 \( 1 + (0.297 + 1.87i)T + (-0.951 + 0.309i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.587 - 0.809i)T^{2} \)
29 \( 1 + (0.734 - 0.533i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.809 - 1.58i)T + (-0.587 + 0.809i)T^{2} \)
41 \( 1 + (-1.87 - 0.610i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.951 - 0.309i)T^{2} \)
53 \( 1 + (-0.183 + 1.16i)T + (-0.951 - 0.309i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.951 + 0.309i)T^{2} \)
71 \( 1 + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.896 + 1.76i)T + (-0.587 - 0.809i)T^{2} \)
79 \( 1 + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.951 + 0.309i)T^{2} \)
89 \( 1 + (-0.550 - 1.69i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.142 + 0.896i)T + (-0.951 - 0.309i)T^{2} \)
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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

−9.897966698847325220363718328858, −9.336045545715528348276387935845, −8.602916110938932053267267472698, −7.55470586932254124357804731271, −6.29509474576031046504658309863, −5.21980349838940910140709753271, −4.67867346131308493841329757364, −3.59585463147861824728990141378, −2.42427285718060389579826026736, −0.968637266781333851888623831607, 2.37899446072060370362761870870, 3.69847492338357884506650748840, 4.25822353329832922846133460837, 5.84006726057176150472361588747, 6.15938622320798448766860038813, 7.25636241112901354279208129536, 7.84908334251719834592495512030, 8.775924941422143080629292112353, 9.677549995403535994008589934281, 10.77715620642285723043289191784

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