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Properties

Label 2-405-9.5-c2-0-0
Degree $2$
Conductor $405$
Sign $-0.766 + 0.642i$
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  + (−1.93 + 1.11i)2-s + (0.5 − 0.866i)4-s + (−1.93 − 1.11i)5-s + (3 + 5.19i)7-s − 6.70i·8-s + 5.00·10-s + (3.87 − 2.23i)11-s + (−8 + 13.8i)13-s + (−11.6 − 6.70i)14-s + (9.5 + 16.4i)16-s + 4.47i·17-s − 2·19-s + (−1.93 + 1.11i)20-s + (−5 + 8.66i)22-s + (−11.6 − 6.70i)23-s + ⋯
L(s)  = 1  + (−0.968 + 0.559i)2-s + (0.125 − 0.216i)4-s + (−0.387 − 0.223i)5-s + (0.428 + 0.742i)7-s − 0.838i·8-s + 0.500·10-s + (0.352 − 0.203i)11-s + (−0.615 + 1.06i)13-s + (−0.829 − 0.479i)14-s + (0.593 + 1.02i)16-s + 0.263i·17-s − 0.105·19-s + (−0.0968 + 0.0559i)20-s + (−0.227 + 0.393i)22-s + (−0.505 − 0.291i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (296, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ -0.766 + 0.642i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0524834 - 0.144197i\)
\(L(\frac12)\) \(\approx\) \(0.0524834 - 0.144197i\)
\(L(2)\) 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 \( 1 + (1.93 + 1.11i)T \)
good2 \( 1 + (1.93 - 1.11i)T + (2 - 3.46i)T^{2} \)
7 \( 1 + (-3 - 5.19i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-3.87 + 2.23i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (8 - 13.8i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 4.47iT - 289T^{2} \)
19 \( 1 + 2T + 361T^{2} \)
23 \( 1 + (11.6 + 6.70i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (27.1 - 15.6i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-9 + 15.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 16T + 1.36e3T^{2} \)
41 \( 1 + (54.2 + 31.3i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (8 + 13.8i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (42.6 - 24.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 4.47iT - 2.80e3T^{2} \)
59 \( 1 + (3.87 + 2.23i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (41 + 71.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (12 - 20.7i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 125. iT - 5.04e3T^{2} \)
73 \( 1 + 74T + 5.32e3T^{2} \)
79 \( 1 + (69 + 119. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (81.3 - 46.9i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 107. iT - 7.92e3T^{2} \)
97 \( 1 + (-83 - 143. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.67844394722651102010894893778, −10.47515963175351484911534819110, −9.394646913540634051056978869999, −8.831056060001859962071768715916, −8.052884517060027916364915472510, −7.14215922317609484793223257465, −6.20089988989085521849222319231, −4.84292658764540987784707405111, −3.66750442383478570843400769290, −1.80995418471933533059442319523, 0.094264536159551397494271509950, 1.51832174230599798408075286416, 3.01380016694650550325465066719, 4.44513144876836018399967967255, 5.57687945345266415416219105165, 7.10925933311462358363762477834, 7.87711990774809287071946415055, 8.668005366602920796262544304131, 9.903480085102080214606449592352, 10.25598700486923443960283562808

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