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Properties

Label 2-405-9.4-c1-0-10
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)4-s + (−0.5 + 0.866i)5-s + (−1 − 1.73i)7-s + (−1.5 − 2.59i)11-s + (2 − 3.46i)13-s + (−1.99 − 3.46i)16-s + 6·17-s − 19-s + (0.999 + 1.73i)20-s + (−3 + 5.19i)23-s + (−0.499 − 0.866i)25-s − 3.99·28-s + (−4.5 − 7.79i)29-s + (0.5 − 0.866i)31-s + 1.99·35-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)4-s + (−0.223 + 0.387i)5-s + (−0.377 − 0.654i)7-s + (−0.452 − 0.783i)11-s + (0.554 − 0.960i)13-s + (−0.499 − 0.866i)16-s + 1.45·17-s − 0.229·19-s + (0.223 + 0.387i)20-s + (−0.625 + 1.08i)23-s + (−0.0999 − 0.173i)25-s − 0.755·28-s + (−0.835 − 1.44i)29-s + (0.0898 − 0.155i)31-s + 0.338·35-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

\(L(1)\) \(\approx\) \(1.02546 - 0.860470i\)
\(L(\frac12)\) \(\approx\) \(1.02546 - 0.860470i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.73i)T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7 - 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 + (-2 - 3.46i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.98682490534469542709071721179, −10.20257874989331194841245820005, −9.591002042220460826834510173618, −8.001586516442057116452193189702, −7.42259803845078450566992005547, −6.02836041327421779299848977799, −5.65446331133247631398115910295, −3.91093827247112766791027217507, −2.78601023189117576833444901547, −0.902664552866366839546307342608, 2.03719337583772445132671745598, 3.33271500820978335412958428860, 4.45589045072803723784453948044, 5.80033750515266137045392067283, 6.86883299896520198180623311240, 7.78568872840796869157171119700, 8.638665127483770727209962501065, 9.510039272189553421049646817551, 10.65747478322144581995655218109, 11.64565840240670253074001782993

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