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Properties

Label 2-30e2-36.31-c0-0-1
Degree $2$
Conductor $900$
Sign $0.939 + 0.342i$
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  + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + 0.999·6-s + (−0.866 + 0.5i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)12-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.499i)18-s − 0.999·21-s + (−0.866 − 0.5i)23-s + (0.499 − 0.866i)24-s + 0.999i·27-s + 0.999i·28-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + 0.999·6-s + (−0.866 + 0.5i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)12-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.499i)18-s − 0.999·21-s + (−0.866 − 0.5i)23-s + (0.499 − 0.866i)24-s + 0.999i·27-s + 0.999i·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(0.449158\)
Root analytic conductor: \(0.670192\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :0),\ 0.939 + 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.911711831\)
\(L(\frac12)\) \(\approx\) \(1.911711831\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
good7 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T + T^{2} \)
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

−10.08910766686788402894711233567, −9.714010054932680819325937152914, −8.807323760400979211374242548428, −7.76459623773891057538114256540, −6.63735453477614437838846371570, −5.79215999053065059993379423144, −4.71981590459968416260113368717, −3.80229274777523275152376534826, −2.96282099632115777682341682216, −2.02958261785064266889205810693, 1.99320744234721752153630222029, 3.29523718287608912861619510852, 3.78266634191057359383353503183, 5.07538116913808543559181474363, 6.30309663225975107875380222692, 6.85346132123599879990300358046, 7.68648435497163247384265529112, 8.420129043678471038269485448510, 9.406643272632738448593183961729, 10.24500534921441458018756521182

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