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Properties

Label 2-30e2-100.39-c0-0-1
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  + (0.587 + 0.809i)2-s + (−0.309 + 0.951i)4-s + (0.951 − 0.309i)5-s + (−0.951 + 0.309i)8-s + (0.809 + 0.587i)10-s + (−1.11 + 1.53i)13-s + (−0.809 − 0.587i)16-s + (1.53 − 0.5i)17-s + i·20-s + (0.809 − 0.587i)25-s − 1.90·26-s + (0.363 − 1.11i)29-s − i·32-s + (1.30 + 0.951i)34-s + (−0.690 + 0.951i)37-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (−0.309 + 0.951i)4-s + (0.951 − 0.309i)5-s + (−0.951 + 0.309i)8-s + (0.809 + 0.587i)10-s + (−1.11 + 1.53i)13-s + (−0.809 − 0.587i)16-s + (1.53 − 0.5i)17-s + i·20-s + (0.809 − 0.587i)25-s − 1.90·26-s + (0.363 − 1.11i)29-s − i·32-s + (1.30 + 0.951i)34-s + (−0.690 + 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: $\chi_{900} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :0),\ 0.187 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.433995820\)
\(L(\frac12)\) \(\approx\) \(1.433995820\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 - 0.809i)T \)
3 \( 1 \)
5 \( 1 + (-0.951 + 0.309i)T \)
good7 \( 1 + T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.309 - 0.951i)T^{2} \)
29 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \)
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 \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \)
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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

−10.17275931231641215449387813048, −9.570579822223241487517302830793, −8.808902022868908360733944534188, −7.80457679397067660606719605163, −6.92742001504528970553325133980, −6.20654542551612198637112975499, −5.18831610434289913067808694988, −4.63597178645526355306897091966, −3.29103757564748139504912281334, −2.02795222061743775303138873349, 1.45088841318367588521960624908, 2.76147638659939290541559603197, 3.43550196516622281907759806418, 5.04980179676393414924804500449, 5.45158749813959260760590138798, 6.42101404924380894289029439744, 7.53603689433988954533114802661, 8.654783942423682096603014717413, 9.807110744934804262501414843221, 10.13286667508176664636120232817

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