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Properties

Label 2-405-9.7-c1-0-13
Degree $2$
Conductor $405$
Sign $-0.173 + 0.984i$
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 − 1.73i)2-s + (−0.999 − 1.73i)4-s + (0.5 + 0.866i)5-s + (1.5 − 2.59i)7-s + 1.99·10-s + (1 − 1.73i)11-s + (2.5 + 4.33i)13-s + (−3 − 5.19i)14-s + (1.99 − 3.46i)16-s − 8·17-s + 19-s + (1 − 1.73i)20-s + (−1.99 − 3.46i)22-s + (−3 − 5.19i)23-s + (−0.499 + 0.866i)25-s + 10·26-s + ⋯
L(s)  = 1  + (0.707 − 1.22i)2-s + (−0.499 − 0.866i)4-s + (0.223 + 0.387i)5-s + (0.566 − 0.981i)7-s + 0.632·10-s + (0.301 − 0.522i)11-s + (0.693 + 1.20i)13-s + (−0.801 − 1.38i)14-s + (0.499 − 0.866i)16-s − 1.94·17-s + 0.229·19-s + (0.223 − 0.387i)20-s + (−0.426 − 0.738i)22-s + (−0.625 − 1.08i)23-s + (−0.0999 + 0.173i)25-s + 1.96·26-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.173 + 0.984i$
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: \((2,\ 405,\ (\ :1/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40333 - 1.67243i\)
\(L(\frac12)\) \(\approx\) \(1.40333 - 1.67243i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.73i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-1.5 + 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 8T + 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 + (-5 - 8.66i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2 - 3.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.5 - 7.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 5T + 73T^{2} \)
79 \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3 - 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.22882763042049182602553103669, −10.55245329301344860005968789240, −9.484172136300199741271880591972, −8.389656835759033156851899294891, −7.09209990093959457095039561185, −6.17554562770879025472472323201, −4.46209447012997208372924027680, −4.13110739194272935515026937616, −2.66895190097342901208957581953, −1.43317395717623672860292060019, 2.02672669427142178442363967082, 3.93138456830584759827346734526, 5.03289313192704901915783090092, 5.71037283355038685346066876541, 6.57678207626383116409346854508, 7.73910683689901688485496081239, 8.489921626606955614305488679115, 9.369790042733689608550413883129, 10.70322080651256461515156567249, 11.64645548910393506153558759244

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