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Properties

Label 2-405-5.3-c2-0-18
Degree $2$
Conductor $405$
Sign $0.980 - 0.198i$
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  + (−2.56 + 2.56i)2-s − 9.18i·4-s + (−3.64 + 3.42i)5-s + (−4.54 + 4.54i)7-s + (13.3 + 13.3i)8-s + (0.573 − 18.1i)10-s + 7.32·11-s + (−13.8 − 13.8i)13-s − 23.3i·14-s − 31.6·16-s + (−20.9 + 20.9i)17-s − 0.814i·19-s + (31.4 + 33.4i)20-s + (−18.8 + 18.8i)22-s + (−15.6 − 15.6i)23-s + ⋯
L(s)  = 1  + (−1.28 + 1.28i)2-s − 2.29i·4-s + (−0.729 + 0.684i)5-s + (−0.648 + 0.648i)7-s + (1.66 + 1.66i)8-s + (0.0573 − 1.81i)10-s + 0.666·11-s + (−1.06 − 1.06i)13-s − 1.66i·14-s − 1.97·16-s + (−1.23 + 1.23i)17-s − 0.0428i·19-s + (1.57 + 1.67i)20-s + (−0.855 + 0.855i)22-s + (−0.682 − 0.682i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.980 - 0.198i$
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: \((2,\ 405,\ (\ :1),\ 0.980 - 0.198i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.331065 + 0.0332555i\)
\(L(\frac12)\) \(\approx\) \(0.331065 + 0.0332555i\)
\(L(2)\) 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 \( 1 + (3.64 - 3.42i)T \)
good2 \( 1 + (2.56 - 2.56i)T - 4iT^{2} \)
7 \( 1 + (4.54 - 4.54i)T - 49iT^{2} \)
11 \( 1 - 7.32T + 121T^{2} \)
13 \( 1 + (13.8 + 13.8i)T + 169iT^{2} \)
17 \( 1 + (20.9 - 20.9i)T - 289iT^{2} \)
19 \( 1 + 0.814iT - 361T^{2} \)
23 \( 1 + (15.6 + 15.6i)T + 529iT^{2} \)
29 \( 1 + 22.9iT - 841T^{2} \)
31 \( 1 - 47.4T + 961T^{2} \)
37 \( 1 + (-11.0 + 11.0i)T - 1.36e3iT^{2} \)
41 \( 1 + 2.80T + 1.68e3T^{2} \)
43 \( 1 + (-41.2 - 41.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (-51.9 + 51.9i)T - 2.20e3iT^{2} \)
53 \( 1 + (-20.9 - 20.9i)T + 2.80e3iT^{2} \)
59 \( 1 - 38.9iT - 3.48e3T^{2} \)
61 \( 1 + 5.73T + 3.72e3T^{2} \)
67 \( 1 + (-54.9 + 54.9i)T - 4.48e3iT^{2} \)
71 \( 1 + 62.1T + 5.04e3T^{2} \)
73 \( 1 + (-8.47 - 8.47i)T + 5.32e3iT^{2} \)
79 \( 1 + 26.5iT - 6.24e3T^{2} \)
83 \( 1 + (51.8 + 51.8i)T + 6.88e3iT^{2} \)
89 \( 1 + 79.7iT - 7.92e3T^{2} \)
97 \( 1 + (-61.7 + 61.7i)T - 9.40e3iT^{2} \)
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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

−10.56947459544085260123570562572, −10.03863887389731548794573269085, −8.998792905629831616721108602896, −8.231163262210774163413321469135, −7.45465822423045713705555800398, −6.45639405853455533839879003915, −5.96399200185558011827855183827, −4.34780267122152649121100348537, −2.53337578011472565974973638678, −0.30102080823318750274862993689, 0.893798655412328903346709409215, 2.40716495194493678668343117892, 3.77039780962208748654927373065, 4.59248371475803517401630860086, 6.85991320883225009265468313308, 7.50467139991002797660123120919, 8.678247189861766499005037499160, 9.337527353822419821377233325019, 9.915800603378614465907063822305, 11.07011450299715023073025616555

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