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Properties

Label 2-405-9.7-c1-0-7
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: $\chi_{405} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.984 + 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92412 - 0.168338i\)
\(L(\frac12)\) \(\approx\) \(1.92412 - 0.168338i\)
\(L(\frac{3}{2})\) 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

−11.16624559026033408609084226763, −10.37791830239912004971699963908, −9.928308657902474119430882027611, −8.035331299489770705053724023178, −7.55711850360958116640063782291, −6.86752763437159531934474806987, −5.05517165695887704667901090559, −4.24866540564882009891139629645, −3.01520316115057184064501069483, −1.67580234953427853447892219728, 1.56852296402028340279410962845, 2.93759745194110355502911529878, 4.98604381478938733878371296454, 5.47812845299257170782773362205, 6.17244297190266264245776517807, 7.67892939728253523283886597651, 8.411473963649291361827553767377, 9.377882143004051183326027534374, 10.41241874163121413025356255186, 11.49101144858109172841332660552

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