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Properties

Label 2-405-27.22-c1-0-6
Degree $2$
Conductor $405$
Sign $0.925 + 0.379i$
Analytic cond. $3.23394$
Root an. cond. $1.79831$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.62 + 1.36i)2-s + (0.432 − 2.45i)4-s + (−0.939 + 0.342i)5-s + (−0.0109 − 0.0622i)7-s + (0.522 + 0.904i)8-s + (1.05 − 1.83i)10-s + (1.08 + 0.395i)11-s + (−4.71 − 3.95i)13-s + (0.102 + 0.0861i)14-s + (2.60 + 0.947i)16-s + (1.03 − 1.79i)17-s + (0.296 + 0.512i)19-s + (0.432 + 2.45i)20-s + (−2.30 + 0.838i)22-s + (1.29 − 7.34i)23-s + ⋯
L(s)  = 1  + (−1.14 + 0.963i)2-s + (0.216 − 1.22i)4-s + (−0.420 + 0.152i)5-s + (−0.00414 − 0.0235i)7-s + (0.184 + 0.319i)8-s + (0.335 − 0.580i)10-s + (0.327 + 0.119i)11-s + (−1.30 − 1.09i)13-s + (0.0274 + 0.0230i)14-s + (0.650 + 0.236i)16-s + (0.251 − 0.435i)17-s + (0.0679 + 0.117i)19-s + (0.0967 + 0.548i)20-s + (−0.491 + 0.178i)22-s + (0.270 − 1.53i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.925 + 0.379i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.925 + 0.379i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.496184 - 0.0978846i\)
\(L(\frac12)\) \(\approx\) \(0.496184 - 0.0978846i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.939 - 0.342i)T \)
good2 \( 1 + (1.62 - 1.36i)T + (0.347 - 1.96i)T^{2} \)
7 \( 1 + (0.0109 + 0.0622i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-1.08 - 0.395i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (4.71 + 3.95i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-1.03 + 1.79i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.296 - 0.512i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.29 + 7.34i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-1.76 + 1.48i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-1.49 + 8.45i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (2.17 - 3.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.08 - 4.27i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-10.3 - 3.77i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.02 + 5.81i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 0.617T + 53T^{2} \)
59 \( 1 + (10.3 - 3.76i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (2.06 + 11.7i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (3.61 + 3.03i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-1.42 + 2.47i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.45 - 2.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.93 - 3.30i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-2.57 + 2.15i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (5.76 + 9.99i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (12.0 + 4.40i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.84643318387527566448036793502, −9.997837160087313792760482511056, −9.325719657538352912072158359851, −8.197739158273999952137905581963, −7.64896326590679708144616386561, −6.79900286010519819584297974735, −5.78083355428124473511898516206, −4.47905939438653467830461960835, −2.79836183844296970095216405361, −0.51596461144070378855184350600, 1.40306976893942502551569179314, 2.76012783872337882577593478991, 4.08203056665920243210017125521, 5.45499964121780083370878935383, 7.05655429112159024110501090272, 7.79392110748525763803912637477, 9.032877600297216469465505727298, 9.325456317674453669361245139803, 10.44223405174832254654688903905, 11.15093428886670813365918182714

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