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Properties

Label 2-405-5.3-c2-0-15
Degree $2$
Conductor $405$
Sign $0.136 - 0.990i$
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  + (−1.18 + 1.18i)2-s + 1.18i·4-s + (−2.02 − 4.57i)5-s + (−1.94 + 1.94i)7-s + (−6.15 − 6.15i)8-s + (7.83 + 3.02i)10-s + 14.5·11-s + (−4.15 − 4.15i)13-s − 4.60i·14-s + 9.88·16-s + (5.66 − 5.66i)17-s + 30.5i·19-s + (5.39 − 2.38i)20-s + (−17.2 + 17.2i)22-s + (15.9 + 15.9i)23-s + ⋯
L(s)  = 1  + (−0.593 + 0.593i)2-s + 0.295i·4-s + (−0.404 − 0.914i)5-s + (−0.277 + 0.277i)7-s + (−0.768 − 0.768i)8-s + (0.783 + 0.302i)10-s + 1.32·11-s + (−0.319 − 0.319i)13-s − 0.329i·14-s + 0.617·16-s + (0.333 − 0.333i)17-s + 1.60i·19-s + (0.269 − 0.119i)20-s + (−0.784 + 0.784i)22-s + (0.695 + 0.695i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.136 - 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.136 - 0.990i$
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: \((2,\ 405,\ (\ :1),\ 0.136 - 0.990i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.774262 + 0.674803i\)
\(L(\frac12)\) \(\approx\) \(0.774262 + 0.674803i\)
\(L(2)\) 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 \( 1 + (2.02 + 4.57i)T \)
good2 \( 1 + (1.18 - 1.18i)T - 4iT^{2} \)
7 \( 1 + (1.94 - 1.94i)T - 49iT^{2} \)
11 \( 1 - 14.5T + 121T^{2} \)
13 \( 1 + (4.15 + 4.15i)T + 169iT^{2} \)
17 \( 1 + (-5.66 + 5.66i)T - 289iT^{2} \)
19 \( 1 - 30.5iT - 361T^{2} \)
23 \( 1 + (-15.9 - 15.9i)T + 529iT^{2} \)
29 \( 1 + 1.74iT - 841T^{2} \)
31 \( 1 - 51.0T + 961T^{2} \)
37 \( 1 + (-2.78 + 2.78i)T - 1.36e3iT^{2} \)
41 \( 1 + 33.1T + 1.68e3T^{2} \)
43 \( 1 + (-28.0 - 28.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (30.1 - 30.1i)T - 2.20e3iT^{2} \)
53 \( 1 + (4.77 + 4.77i)T + 2.80e3iT^{2} \)
59 \( 1 - 80.6iT - 3.48e3T^{2} \)
61 \( 1 - 15.2T + 3.72e3T^{2} \)
67 \( 1 + (-45.2 + 45.2i)T - 4.48e3iT^{2} \)
71 \( 1 - 45.4T + 5.04e3T^{2} \)
73 \( 1 + (-92.0 - 92.0i)T + 5.32e3iT^{2} \)
79 \( 1 - 32.5iT - 6.24e3T^{2} \)
83 \( 1 + (4.89 + 4.89i)T + 6.88e3iT^{2} \)
89 \( 1 + 35.7iT - 7.92e3T^{2} \)
97 \( 1 + (-43.7 + 43.7i)T - 9.40e3iT^{2} \)
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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

−11.49974073076185244985950351460, −9.841301940219842422570891310285, −9.325686918391978367735561177982, −8.387929101597314147021672517459, −7.76539560767953242716531805516, −6.68414204795340173977226700482, −5.69712834961132384820634669263, −4.29295488432858446824368993339, −3.26496043012306110817266097211, −1.09956043500720577362269823343, 0.68785889848348748128646283086, 2.32736344084827804250528832729, 3.49449770396141452798228737294, 4.84168647311413629508193583279, 6.45880442773252901857021910390, 6.85692241590686854539899777427, 8.308611977279579224816275115509, 9.230172701947112147421696496043, 9.998401364016051847901708002500, 10.81380556968031122567564366227

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