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Properties

Label 2-405-15.14-c2-0-35
Degree $2$
Conductor $405$
Sign $0.948 + 0.318i$
Analytic cond. $11.0354$
Root an. cond. $3.32196$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.68·2-s + 9.59·4-s + (1.59 − 4.74i)5-s + 2.56i·7-s + 20.6·8-s + (5.86 − 17.4i)10-s + 9.83i·11-s − 12.0i·13-s + 9.46i·14-s + 37.6·16-s − 4.28·17-s + 7.16·19-s + (15.2 − 45.4i)20-s + 36.2i·22-s − 0.510·23-s + ⋯
L(s)  = 1  + 1.84·2-s + 2.39·4-s + (0.318 − 0.948i)5-s + 0.366i·7-s + 2.57·8-s + (0.586 − 1.74i)10-s + 0.893i·11-s − 0.927i·13-s + 0.676i·14-s + 2.35·16-s − 0.252·17-s + 0.377·19-s + (0.763 − 2.27i)20-s + 1.64i·22-s − 0.0221·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.318i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.948 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.948 + 0.318i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ 0.948 + 0.318i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(5.25151 - 0.857905i\)
\(L(\frac12)\) \(\approx\) \(5.25151 - 0.857905i\)
\(L(2)\) 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 \( 1 + (-1.59 + 4.74i)T \)
good2 \( 1 - 3.68T + 4T^{2} \)
7 \( 1 - 2.56iT - 49T^{2} \)
11 \( 1 - 9.83iT - 121T^{2} \)
13 \( 1 + 12.0iT - 169T^{2} \)
17 \( 1 + 4.28T + 289T^{2} \)
19 \( 1 - 7.16T + 361T^{2} \)
23 \( 1 + 0.510T + 529T^{2} \)
29 \( 1 - 30.5iT - 841T^{2} \)
31 \( 1 + 19.2T + 961T^{2} \)
37 \( 1 + 1.31iT - 1.36e3T^{2} \)
41 \( 1 + 34.6iT - 1.68e3T^{2} \)
43 \( 1 - 51.9iT - 1.84e3T^{2} \)
47 \( 1 + 50.9T + 2.20e3T^{2} \)
53 \( 1 + 86.6T + 2.80e3T^{2} \)
59 \( 1 - 105. iT - 3.48e3T^{2} \)
61 \( 1 - 31.3T + 3.72e3T^{2} \)
67 \( 1 + 78.0iT - 4.48e3T^{2} \)
71 \( 1 - 72.6iT - 5.04e3T^{2} \)
73 \( 1 - 30.3iT - 5.32e3T^{2} \)
79 \( 1 - 115.T + 6.24e3T^{2} \)
83 \( 1 + 60.0T + 6.88e3T^{2} \)
89 \( 1 + 71.2iT - 7.92e3T^{2} \)
97 \( 1 + 127. iT - 9.40e3T^{2} \)
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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

−11.39220427653274344550885193418, −10.36429920493520194796715972602, −9.242547045849819024240291748360, −7.943809167059484662288337051650, −6.89319728851444304251867520196, −5.75192423090819268379741953907, −5.13506914800986776726251116918, −4.27317896572181369586060145193, −2.98905741294285895252390941403, −1.71389956711958840388915670158, 2.03671108605870892619303145513, 3.20618409368630654751726868741, 4.03227031422421042122898299567, 5.24487824095010470604072373724, 6.26336118364250290982242628516, 6.80050241148512740747774044396, 7.86704045342001323760728431031, 9.524114319449731377949306648275, 10.71578797840046561306487203778, 11.28959759032676370094647950818

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