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Properties

Label 2-30e2-100.91-c0-0-0
Degree $2$
Conductor $900$
Sign $-0.187 - 0.982i$
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  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)5-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)10-s + (−0.5 + 0.363i)13-s + (−0.809 + 0.587i)16-s + (0.5 − 1.53i)17-s − 20-s + (−0.809 − 0.587i)25-s − 0.618·26-s + (0.5 + 1.53i)29-s − 32-s + (1.30 − 0.951i)34-s + (1.30 − 0.951i)37-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)5-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)10-s + (−0.5 + 0.363i)13-s + (−0.809 + 0.587i)16-s + (0.5 − 1.53i)17-s − 20-s + (−0.809 − 0.587i)25-s − 0.618·26-s + (0.5 + 1.53i)29-s − 32-s + (1.30 − 0.951i)34-s + (1.30 − 0.951i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.187 - 0.982i$
Analytic conductor: \(0.449158\)
Root analytic conductor: \(0.670192\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :0),\ -0.187 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.423963161\)
\(L(\frac12)\) \(\approx\) \(1.423963161\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 \)
5 \( 1 + (0.309 - 0.951i)T \)
good7 \( 1 - T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.809 + 0.587i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
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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

−10.76486800999752715139386487573, −9.683651154086042175510409792962, −8.735601538043212685785742143112, −7.51037078062743494332919133313, −7.24157482103488924726725352968, −6.29984054906616827132096526159, −5.30764636196956021626146305528, −4.37427395437747443119762081843, −3.28030407126397008424340129123, −2.47470684055183168209442359936, 1.23149490184701595948393562657, 2.61240193002872695376815841275, 3.91436252679634624073741527626, 4.55234985015706056675924116951, 5.59659764838152895583636657785, 6.26717098598768207066738525179, 7.63171877349717513785500929750, 8.385713789739642149276021526797, 9.500281353636004488824856649603, 10.13110540383176579835939850246

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