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Properties

Label 2-405-135.122-c1-0-8
Degree $2$
Conductor $405$
Sign $0.0953 - 0.995i$
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  + (1.66 + 1.16i)2-s + (0.729 + 2.00i)4-s + (1.48 + 1.67i)5-s + (1.13 + 2.43i)7-s + (−0.0700 + 0.261i)8-s + (0.518 + 4.51i)10-s + (−4.08 − 4.86i)11-s + (1.19 + 1.70i)13-s + (−0.948 + 5.37i)14-s + (2.84 − 2.38i)16-s + (0.174 + 0.649i)17-s + (−2.70 + 1.56i)19-s + (−2.27 + 4.19i)20-s + (−1.12 − 12.8i)22-s + (0.409 + 0.191i)23-s + ⋯
L(s)  = 1  + (1.17 + 0.824i)2-s + (0.364 + 1.00i)4-s + (0.663 + 0.748i)5-s + (0.429 + 0.920i)7-s + (−0.0247 + 0.0923i)8-s + (0.164 + 1.42i)10-s + (−1.23 − 1.46i)11-s + (0.331 + 0.473i)13-s + (−0.253 + 1.43i)14-s + (0.711 − 0.597i)16-s + (0.0422 + 0.157i)17-s + (−0.620 + 0.358i)19-s + (−0.508 + 0.937i)20-s + (−0.240 − 2.74i)22-s + (0.0854 + 0.0398i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0953 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0953 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.0953 - 0.995i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.0953 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00310 + 1.82032i\)
\(L(\frac12)\) \(\approx\) \(2.00310 + 1.82032i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 - 1.67i)T \)
good2 \( 1 + (-1.66 - 1.16i)T + (0.684 + 1.87i)T^{2} \)
7 \( 1 + (-1.13 - 2.43i)T + (-4.49 + 5.36i)T^{2} \)
11 \( 1 + (4.08 + 4.86i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-1.19 - 1.70i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (-0.174 - 0.649i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.70 - 1.56i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.409 - 0.191i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (1.39 + 7.90i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (1.53 - 0.559i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-2.76 + 0.739i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-9.28 - 1.63i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.307 + 3.51i)T + (-42.3 - 7.46i)T^{2} \)
47 \( 1 + (5.68 - 2.65i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (5.99 + 5.99i)T + 53iT^{2} \)
59 \( 1 + (6.07 + 5.10i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.53 + 0.560i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-4.38 + 3.06i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (2.05 + 1.18i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.33 + 0.358i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (6.73 - 1.18i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-2.27 + 3.24i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (0.428 + 0.742i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.88 + 0.340i)T + (95.5 + 16.8i)T^{2} \)
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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

−11.48941053401956230713612079873, −10.79271864268555366260913848830, −9.656265235087625947452781936425, −8.428264530872101092170747653370, −7.58175396760019496850308580643, −6.11680770324423795359210544406, −5.99352587554852119388312736946, −4.94604887047992735228436144216, −3.52632999693470516695785465688, −2.39608286884260386612457912098, 1.56582141398329910624058420975, 2.74941314293946679341877037523, 4.31762267264958240004996150240, 4.84765429072214487407477604923, 5.76435135301709304724120980209, 7.26619950728636933173933193477, 8.228052902265933913169867025735, 9.547965099151321966670634532467, 10.52600909743428555459876341486, 10.95428624134566872069612492495

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