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Properties

Label 2-405-9.4-c1-0-3
Degree $2$
Conductor $405$
Sign $0.984 - 0.173i$
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  + (−0.366 − 0.633i)2-s + (0.732 − 1.26i)4-s + (−0.5 + 0.866i)5-s + (2.36 + 4.09i)7-s − 2.53·8-s + 0.732·10-s + (2.86 + 4.96i)11-s + (−0.732 + 1.26i)13-s + (1.73 − 3i)14-s + (−0.535 − 0.928i)16-s − 2.73·17-s + 4.46·19-s + (0.732 + 1.26i)20-s + (2.09 − 3.63i)22-s + (1.73 − 3i)23-s + ⋯
L(s)  = 1  + (−0.258 − 0.448i)2-s + (0.366 − 0.633i)4-s + (−0.223 + 0.387i)5-s + (0.894 + 1.54i)7-s − 0.896·8-s + 0.231·10-s + (0.864 + 1.49i)11-s + (−0.203 + 0.351i)13-s + (0.462 − 0.801i)14-s + (−0.133 − 0.232i)16-s − 0.662·17-s + 1.02·19-s + (0.163 + 0.283i)20-s + (0.447 − 0.774i)22-s + (0.361 − 0.625i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34287 + 0.117486i\)
\(L(\frac12)\) \(\approx\) \(1.34287 + 0.117486i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.366 + 0.633i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-2.36 - 4.09i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.86 - 4.96i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.732 - 1.26i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.73T + 17T^{2} \)
19 \( 1 - 4.46T + 19T^{2} \)
23 \( 1 + (-1.73 + 3i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.59 + 2.76i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.73T + 37T^{2} \)
41 \( 1 + (-3.59 + 6.23i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0980 + 0.169i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.36 - 7.56i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.73T + 53T^{2} \)
59 \( 1 + (-4.13 + 7.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.73 - 3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.73T + 71T^{2} \)
73 \( 1 + 7.66T + 73T^{2} \)
79 \( 1 + (7.73 + 13.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.09 + 1.90i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.19T + 89T^{2} \)
97 \( 1 + (-4.83 - 8.36i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.53061023993282105443458891742, −10.41387427268553587915732248925, −9.389930527715667877384248154295, −8.920066587151520944654138723853, −7.52081781168786211895604479539, −6.54677493187342236951340282122, −5.50937241325768617849831822555, −4.45707098216171995792816046018, −2.58646768796043826120598385252, −1.78341626162607574448489394371, 1.06781007769637193604601039689, 3.27690572201446400781601964000, 4.16539898078527563276244983248, 5.53411138505939342050733129579, 6.85076901694255310397055641949, 7.50456106190513090222657709631, 8.358456598159535712604206707716, 9.071732631947064442978278181325, 10.47427028693480789911715830912, 11.38176000394801369503691463226

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