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Properties

Label 2-405-15.14-c2-0-14
Degree $2$
Conductor $405$
Sign $-0.217 - 0.976i$
Analytic cond. $11.0354$
Root an. cond. $3.32196$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.28·2-s + 1.21·4-s + (1.08 + 4.88i)5-s + 3.44i·7-s − 6.35·8-s + (2.48 + 11.1i)10-s + 4.50i·11-s + 11.8i·13-s + 7.87i·14-s − 19.3·16-s − 23.3·17-s + 11.0·19-s + (1.32 + 5.93i)20-s + 10.2i·22-s + 29.8·23-s + ⋯
L(s)  = 1  + 1.14·2-s + 0.304·4-s + (0.217 + 0.976i)5-s + 0.492i·7-s − 0.794·8-s + (0.248 + 1.11i)10-s + 0.409i·11-s + 0.911i·13-s + 0.562i·14-s − 1.21·16-s − 1.37·17-s + 0.582·19-s + (0.0661 + 0.296i)20-s + 0.467i·22-s + 1.29·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 - 0.976i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.217 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.217 - 0.976i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ -0.217 - 0.976i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.47187 + 1.83617i\)
\(L(\frac12)\) \(\approx\) \(1.47187 + 1.83617i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-1.08 - 4.88i)T \)
good2 \( 1 - 2.28T + 4T^{2} \)
7 \( 1 - 3.44iT - 49T^{2} \)
11 \( 1 - 4.50iT - 121T^{2} \)
13 \( 1 - 11.8iT - 169T^{2} \)
17 \( 1 + 23.3T + 289T^{2} \)
19 \( 1 - 11.0T + 361T^{2} \)
23 \( 1 - 29.8T + 529T^{2} \)
29 \( 1 - 35.6iT - 841T^{2} \)
31 \( 1 - 30.3T + 961T^{2} \)
37 \( 1 - 5.11iT - 1.36e3T^{2} \)
41 \( 1 - 22.6iT - 1.68e3T^{2} \)
43 \( 1 + 67.9iT - 1.84e3T^{2} \)
47 \( 1 + 46.2T + 2.20e3T^{2} \)
53 \( 1 - 68.0T + 2.80e3T^{2} \)
59 \( 1 - 34.5iT - 3.48e3T^{2} \)
61 \( 1 - 17.3T + 3.72e3T^{2} \)
67 \( 1 + 98.2iT - 4.48e3T^{2} \)
71 \( 1 - 134. iT - 5.04e3T^{2} \)
73 \( 1 + 134. iT - 5.32e3T^{2} \)
79 \( 1 - 49.3T + 6.24e3T^{2} \)
83 \( 1 - 28.6T + 6.88e3T^{2} \)
89 \( 1 + 65.9iT - 7.92e3T^{2} \)
97 \( 1 + 71.7iT - 9.40e3T^{2} \)
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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

−11.52815962914627151034181186240, −10.64032093700739001949606129654, −9.419814507594721100223810632355, −8.737571769643264022718118656886, −7.05131986469309706278681097911, −6.54417733608901960900914203537, −5.38970409454338613522903762599, −4.45491141990020569837984084918, −3.26545808767988669080104486124, −2.24650170586725800431080041131, 0.69995585166547748896779391515, 2.71532469168692636843454442472, 4.01272222656809059341222973152, 4.85019324134722889445090156866, 5.66974018810080406600425695035, 6.68716642594187580844616319610, 8.073661013131707281965628130800, 8.934185630209246159133105427073, 9.829336878277164315765212760799, 11.06611703245944991620102255072

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