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Properties

Label 2-405-45.29-c2-0-42
Degree $2$
Conductor $405$
Sign $-0.770 - 0.637i$
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  + (1.74 − 3.01i)2-s + (−4.06 − 7.04i)4-s + (3.55 + 3.51i)5-s + (−11.0 − 6.38i)7-s − 14.4·8-s + (16.7 − 4.62i)10-s + (−12.2 − 7.09i)11-s + (3.89 − 2.25i)13-s + (−38.5 + 22.2i)14-s + (−8.83 + 15.3i)16-s + 9.20·17-s − 15.8·19-s + (10.2 − 39.3i)20-s + (−42.8 + 24.7i)22-s + (−2.12 − 3.67i)23-s + ⋯
L(s)  = 1  + (0.870 − 1.50i)2-s + (−1.01 − 1.76i)4-s + (0.711 + 0.702i)5-s + (−1.58 − 0.912i)7-s − 1.80·8-s + (1.67 − 0.462i)10-s + (−1.11 − 0.644i)11-s + (0.299 − 0.173i)13-s + (−2.75 + 1.58i)14-s + (−0.552 + 0.956i)16-s + 0.541·17-s − 0.833·19-s + (0.513 − 1.96i)20-s + (−1.94 + 1.12i)22-s + (−0.0922 − 0.159i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.770 - 0.637i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ -0.770 - 0.637i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.529168 + 1.46916i\)
\(L(\frac12)\) \(\approx\) \(0.529168 + 1.46916i\)
\(L(2)\) 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 \( 1 + (-3.55 - 3.51i)T \)
good2 \( 1 + (-1.74 + 3.01i)T + (-2 - 3.46i)T^{2} \)
7 \( 1 + (11.0 + 6.38i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (12.2 + 7.09i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-3.89 + 2.25i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 9.20T + 289T^{2} \)
19 \( 1 + 15.8T + 361T^{2} \)
23 \( 1 + (2.12 + 3.67i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (22.9 + 13.2i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (15.7 + 27.2i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 14.6iT - 1.36e3T^{2} \)
41 \( 1 + (-38.6 + 22.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-56.6 - 32.6i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-28.2 + 48.9i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 17.8T + 2.80e3T^{2} \)
59 \( 1 + (41.8 - 24.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-45.7 + 79.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-43.8 + 25.3i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 67.7iT - 5.04e3T^{2} \)
73 \( 1 - 52.6iT - 5.32e3T^{2} \)
79 \( 1 + (-27.4 + 47.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-1.27 + 2.21i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 82.7iT - 7.92e3T^{2} \)
97 \( 1 + (-55.3 - 31.9i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.63108208165204202388271396340, −10.01622094598552736149265799900, −9.304184160490782747085152507638, −7.55123810147637224734594538309, −6.24699622491037760349193273384, −5.58144032758470522374707510728, −4.00291068846191814204682471860, −3.21519656618690040494528342790, −2.33959283354642094551032532438, −0.47722890147102395898790779486, 2.58291442416487439753732554839, 3.98866856958648909499217578164, 5.29394628617105097016379580796, 5.78069193874155976701647685913, 6.62105504753593425015674686888, 7.63776138698648071082864586235, 8.778009630336848157832924681230, 9.382407835535472227442060397439, 10.46176832030923340485612853396, 12.38911523179283198794272569264

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