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Properties

Label 2-405-9.4-c1-0-4
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: $\chi_{405} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.753278 - 0.632075i\)
\(L(\frac12)\) \(\approx\) \(0.753278 - 0.632075i\)
\(L(\frac{3}{2})\) 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

−10.88453421847762944602818442222, −10.40718919084505880155866124730, −9.399007788310623279621926700885, −8.363861304114176456682390273130, −7.916929550795261082861775466940, −6.14692839092709099640451756561, −5.25076995185891532626547408924, −3.38933942126719167586635050708, −2.71398954237525408988619170929, −1.08343691321948729009881248607, 1.22103081696222375908404614357, 3.62444156160475202883747639614, 4.89241394723324375503901754068, 5.91061328738847037278892608523, 7.18331000262169702589735157948, 7.58408491165739626257950173207, 8.486703952539787754181347501365, 9.437266625424820203166980756257, 10.21935532222784168167537847357, 11.43889871059080265674462485987

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