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Properties

Label 2-405-5.3-c2-0-43
Degree $2$
Conductor $405$
Sign $-0.635 - 0.771i$
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  + (2.41 − 2.41i)2-s − 7.68i·4-s + (−4.73 − 1.61i)5-s + (−4.71 + 4.71i)7-s + (−8.91 − 8.91i)8-s + (−15.3 + 7.54i)10-s − 14.2·11-s + (−5.12 − 5.12i)13-s + 22.7i·14-s − 12.3·16-s + (−2.45 + 2.45i)17-s − 10.6i·19-s + (−12.3 + 36.3i)20-s + (−34.3 + 34.3i)22-s + (−22.6 − 22.6i)23-s + ⋯
L(s)  = 1  + (1.20 − 1.20i)2-s − 1.92i·4-s + (−0.946 − 0.322i)5-s + (−0.673 + 0.673i)7-s + (−1.11 − 1.11i)8-s + (−1.53 + 0.754i)10-s − 1.29·11-s + (−0.393 − 0.393i)13-s + 1.62i·14-s − 0.771·16-s + (−0.144 + 0.144i)17-s − 0.561i·19-s + (−0.619 + 1.81i)20-s + (−1.56 + 1.56i)22-s + (−0.984 − 0.984i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 - 0.771i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.635 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.635 - 0.771i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ -0.635 - 0.771i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.444155 + 0.941214i\)
\(L(\frac12)\) \(\approx\) \(0.444155 + 0.941214i\)
\(L(2)\) 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 \( 1 + (4.73 + 1.61i)T \)
good2 \( 1 + (-2.41 + 2.41i)T - 4iT^{2} \)
7 \( 1 + (4.71 - 4.71i)T - 49iT^{2} \)
11 \( 1 + 14.2T + 121T^{2} \)
13 \( 1 + (5.12 + 5.12i)T + 169iT^{2} \)
17 \( 1 + (2.45 - 2.45i)T - 289iT^{2} \)
19 \( 1 + 10.6iT - 361T^{2} \)
23 \( 1 + (22.6 + 22.6i)T + 529iT^{2} \)
29 \( 1 + 18.0iT - 841T^{2} \)
31 \( 1 - 30.3T + 961T^{2} \)
37 \( 1 + (8.20 - 8.20i)T - 1.36e3iT^{2} \)
41 \( 1 + 30.3T + 1.68e3T^{2} \)
43 \( 1 + (5.09 + 5.09i)T + 1.84e3iT^{2} \)
47 \( 1 + (-48.8 + 48.8i)T - 2.20e3iT^{2} \)
53 \( 1 + (18.4 + 18.4i)T + 2.80e3iT^{2} \)
59 \( 1 - 6.64iT - 3.48e3T^{2} \)
61 \( 1 + 71.2T + 3.72e3T^{2} \)
67 \( 1 + (27.5 - 27.5i)T - 4.48e3iT^{2} \)
71 \( 1 - 25.2T + 5.04e3T^{2} \)
73 \( 1 + (14.2 + 14.2i)T + 5.32e3iT^{2} \)
79 \( 1 + 43.7iT - 6.24e3T^{2} \)
83 \( 1 + (42.6 + 42.6i)T + 6.88e3iT^{2} \)
89 \( 1 + 165. iT - 7.92e3T^{2} \)
97 \( 1 + (-5.16 + 5.16i)T - 9.40e3iT^{2} \)
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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

−10.61010443844361972333784865938, −10.05620303734619620173488827695, −8.729756002193411397985193446935, −7.71215227239343650037888647246, −6.22769729842444020060001106695, −5.17879358421085622330044144395, −4.38814701372821401780665070147, −3.18070851704572302769219124365, −2.38222421358348954663774019658, −0.28391722155259050348122563308, 2.99963614916424988560956497366, 3.91744059278987877910468935707, 4.82479730988507178650661964092, 5.94042175880345178145295836358, 6.95833502571409303766449255406, 7.57118394247513960979638001224, 8.277509852734849197472575840557, 9.871472471598803715296727467483, 10.83158303211725469963122636488, 12.04341216224716702314774285851

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